Calculus Test 1 Review Guide – Step-by-Step Study Guidance

{"type":"doc","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"{\"type\":\"doc\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q1. The height of a baseball can be found by using the formula \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(t) = -16t^2 + 60t + 6\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\". Evaluate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1.5)\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", then find the average velocity of the baseball over the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"[1, 1.5]\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Functions and Average Rate of Change\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate a function at specific points and calculate the average velocity (rate of change) over an interval.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(t)\"}},{\"type\":\"text\",\"text\":\": Position function (height as a function of time)\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Average velocity over \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"[a, b]\"}},{\"type\":\"text\",\"text\":\": \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{s(b) - s(a)}{b - a}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1)\"}},{\"type\":\"text\",\"text\":\" by plugging \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = 1\"}},{\"type\":\"text\",\"text\":\" into the formula: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1) = -16(1)^2 + 60(1) + 6\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1.5)\"}},{\"type\":\"text\",\"text\":\" by plugging \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"t = 1.5\"}},{\"type\":\"text\",\"text\":\" into the formula: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1.5) = -16(1.5)^2 + 60(1.5) + 6\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Set up the average velocity formula: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\text{Average velocity} = \\\\frac{s(1.5) - s(1)}{1.5 - 1}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1)\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1.5)\"}},{\"type\":\"text\",\"text\":\", then substitute these values into the average velocity formula.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1) = -16(1)^2 + 60(1) + 6 = -16 + 60 + 6 = 50\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"s(1.5) = -16(1.5)^2 + 60(1.5) + 6 = -16(2.25) + 90 + 6 = -36 + 90 + 6 = 60\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Average velocity: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{60 - 50}{1.5 - 1} = \\\\frac{10}{0.5} = 20\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The average velocity of the baseball over \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"[1, 1.5]\"}},{\"type\":\"text\",\"text\":\" is $20"},{"type":"inlineMath","attrs":{"latex":" ft/s.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q2. Use the graph to evaluate:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"a. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(1)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"b. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1^-} f(x)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"c. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1^+} f(x)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"d. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1} f(x)\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits and Function Values from Graphs\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to read function values and limits from a graph, including one-sided limits.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(1)\"}},{\"type\":\"text\",\"text\":\": The value of the function at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 1\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1^-} f(x)\"}},{\"type\":\"text\",\"text\":\": Limit as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches $1"}},{"type":"text","text":" from the left\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1^+} f(x)\"}},{\"type\":\"text\",\"text\":\": Limit as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches $1"},{"type":"inlineMath","attrs":{"latex":" from the right\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1} f(x)\"}},{\"type\":\"text\",\"text\":\": Limit as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches $1"}},{"type":"text","text":" (exists if left and right limits are equal)\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Look at the graph and find the value of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(1)\"}},{\"type\":\"text\",\"text\":\" (the point at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 1\"}},{\"type\":\"text\",\"text\":\").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Observe the behavior of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches "},{"type":"inlineMath","attrs":{"latex":"1\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" from the left (\"}},{\"type\":\"text\",\"text\":\"x < 1"}},{"type":"text","text":").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Observe the behavior of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x)\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches "},{"type":"inlineMath","attrs":{"latex":"1\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" from the right (\"}},{\"type\":\"text\",\"text\":\"x > 1"}},{"type":"text","text":").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compare the left and right limits to determine if the overall limit exists at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 1\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"a. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(1)\"}},{\"type\":\"text\",\"text\":\": (Value from graph)\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"b. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1^-} f(x)\"}},{\"type\":\"text\",\"text\":\": (Left-hand limit from graph)\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"c. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1^+} f(x)\"}},{\"type\":\"text\",\"text\":\": (Right-hand limit from graph)\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"d. \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 1} f(x)\"}},{\"type\":\"text\",\"text\":\": (Exists if left and right limits are equal; otherwise, does not exist)\"}]}]}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check the graph for the exact values.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q3. Use the graph to evaluate the function on the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"[-1, 4]\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", determine at which values of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" the function fails to be continuous.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Continuity from Graphs\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to identify points of discontinuity by examining a graph.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Continuous function: No breaks, jumps, or holes in the graph.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Discontinuity: A point where the function is not continuous.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Scan the graph over the interval \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"[-1, 4]\"}},{\"type\":\"text\",\"text\":\" for any jumps, holes, or vertical asymptotes.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"List the \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\"-values where the function is not continuous.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check if the function is defined at those points and if the limit exists.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The function fails to be continuous at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = \"}},{\"type\":\"text\",\"text\":\" (list all points of discontinuity found on the graph).\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Discontinuities can be due to jumps, holes, or vertical asymptotes.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q4. Estimate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 0} \\\\frac{\\\\sin 2x}{\\\\sin x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" by graphing.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits and Graphical Estimation\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to estimate a limit using a graph, especially when the expression is indeterminate at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin x\"}},{\"type\":\"text\",\"text\":\": Sine function\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Indeterminate form: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{0}{0}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Graph \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"y = \\\\frac{\\\\sin 2x}{\\\\sin x}\"}},{\"type\":\"text\",\"text\":\" near \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 0\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Observe the behavior of the function as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches $0"},{"type":"inlineMath","attrs":{"latex":" from both sides.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Estimate the value the function approaches as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" gets closer to $0"}},{"type":"text","text":".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 0} \\\\frac{\\\\sin 2x}{\\\\sin x} = 2\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The graph shows the function approaches "},{"type":"inlineMath","attrs":{"latex":"2\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" as \"}},{\"type\":\"text\",\"text\":\"x\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" approaches $0\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q5. Analytically find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 0} \\\\frac{\\\\sin 2x}{\\\\sin x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", using the trigonometric identity \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin 2x = 2 \\\\sin x \\\\cos x\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits and Trigonometric Identities\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to use identities to simplify a limit and evaluate it analytically.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin 2x = 2 \\\\sin x \\\\cos x\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Limit properties\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Rewrite \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{\\\\sin 2x}{\\\\sin x}\"}},{\"type\":\"text\",\"text\":\" using the identity: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{2 \\\\sin x \\\\cos x}{\\\\sin x}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify the expression by canceling \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin x\"}},{\"type\":\"text\",\"text\":\" (for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\neq 0\"}},{\"type\":\"text\",\"text\":\").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate the limit as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to 0\"}},{\"type\":\"text\",\"text\":\" of the simplified expression.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 0} \\\\frac{\\\\sin 2x}{\\\\sin x} = \\\\lim_{x \\\\to 0} 2 \\\\cos x = 2\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches $0\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\", \"}},{\"type\":\"text\",\"text\":\"\\\\cos x\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" approaches $1\"}},{\"type\":\"text\",\"text\":\", so the limit is $2"}},{"type":"text","text":".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q6. Estimate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\frac{\\\\pi}{4}} \\\\frac{\\\\cos 2x}{\\\\cos x - \\\\sin x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" by making a table (round to 4 digits).\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits and Numerical Estimation\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to estimate a limit numerically by evaluating the function at values close to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = \\\\frac{\\\\pi}{4}\"}},{\"type\":\"text\",\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Table of values: Plug in values near \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = \\\\frac{\\\\pi}{4}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Round answers to 4 decimal places\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Choose values of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaching \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{\\\\pi}{4}\"}},{\"type\":\"text\",\"text\":\" from both sides (e.g., \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"0.75\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"0.77\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"0.78\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"0.79\"}},{\"type\":\"text\",\"text\":\" radians).\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos 2x\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos x - \\\\sin x\"}},{\"type\":\"text\",\"text\":\" for each value.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Compute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{\\\\cos 2x}{\\\\cos x - \\\\sin x}\"}},{\"type\":\"text\",\"text\":\" for each value and observe the trend.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\frac{\\\\pi}{4}} \\\\frac{\\\\cos 2x}{\\\\cos x - \\\\sin x} = -2.0000\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Numerical estimation shows the limit approaches \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-2\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" gets closer to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{\\\\pi}{4}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q7. Analytically find \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\frac{\\\\pi}{4}} \\\\frac{\\\\cos 2x}{\\\\cos x - \\\\sin x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Analytical Limits and Trigonometric Identities\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to use trigonometric identities and algebra to evaluate a limit analytically.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos 2x = 1 - 2 \\\\sin^2 x\"}},{\"type\":\"text\",\"text\":\" or \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos 2x = 2 \\\\cos^2 x - 1\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate denominator at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = \\\\frac{\\\\pi}{4}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = \\\\frac{\\\\pi}{4}\"}},{\"type\":\"text\",\"text\":\" into \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos 2x\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos x - \\\\sin x\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\frac{\\\\pi}{4}\"}},{\"type\":\"text\",\"text\":\" and \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sin \\\\frac{\\\\pi}{4}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Plug these values into the expression and simplify.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\frac{\\\\pi}{4}} \\\\frac{\\\\cos 2x}{\\\\cos x - \\\\sin x} = -2\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Substituting the values gives \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos 2x = \\\\cos \\\\frac{\\\\pi}{2} = 0\"}},{\"type\":\"text\",\"text\":\", denominator \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos \\\\frac{\\\\pi}{4} - \\\\sin \\\\frac{\\\\pi}{4} = 0\"}},{\"type\":\"text\",\"text\":\", but using limits and identities, the result is \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-2\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q8. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{h \\\\to 0} \\\\frac{\\\\sqrt{5x + h} - \\\\sqrt{5x}}{h}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\", where \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x > 0\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" is a constant.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Difference Quotient and Derivatives\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate a limit that resembles the definition of the derivative.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Difference quotient: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{f(x + h) - f(x)}{h}\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Derivative: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{h \\\\to 0} \\\\frac{f(x + h) - f(x)}{h}\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recognize the expression as the difference quotient for \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"f(x) = \\\\sqrt{5x}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply numerator and denominator by the conjugate: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sqrt{5x + h} + \\\\sqrt{5x}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Simplify the numerator using the difference of squares.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cancel \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"h\"}},{\"type\":\"text\",\"text\":\" and take the limit as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"h \\\\to 0\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{h \\\\to 0} \\\\frac{\\\\sqrt{5x + h} - \\\\sqrt{5x}}{h} = \\\\frac{1}{2 \\\\sqrt{5x}}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This is the derivative of \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\sqrt{5x}\"}},{\"type\":\"text\",\"text\":\" with respect to \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q9. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{p \\\\to 1} \\\\frac{p^5 - 1}{p - 1}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits and Factoring\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate a limit by factoring and canceling terms.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"p^5 - 1\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(p - 1)(p^4 + p^3 + p^2 + p + 1)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cancel common factors\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor the numerator: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"p^5 - 1 = (p - 1)(p^4 + p^3 + p^2 + p + 1)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cancel \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(p - 1)\"}},{\"type\":\"text\",\"text\":\" in numerator and denominator.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate the remaining expression at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"p = 1\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{p \\\\to 1} \\\\frac{p^5 - 1}{p - 1} = 5\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"After canceling, substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"p = 1\"}},{\"type\":\"text\",\"text\":\" into \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"p^4 + p^3 + p^2 + p + 1\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q10. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 3} \\\\frac{x^4 - 81}{x - 3}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits and Factoring\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to factor polynomials and evaluate limits.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^4 - 81\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x - 3)(x^3 + 3x^2 + 9x + 27)\"}}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cancel common factors\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor the numerator: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^4 - 81 = (x - 3)(x^3 + 3x^2 + 9x + 27)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Cancel \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"(x - 3)\"}},{\"type\":\"text\",\"text\":\" in numerator and denominator.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate the remaining expression at \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 3\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 3} \\\\frac{x^4 - 81}{x - 3} = 60\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"After canceling, substitute \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 3\"}},{\"type\":\"text\",\"text\":\" into \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^3 + 3x^2 + 9x + 27\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q11. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 3^-} \\\\frac{x - 4}{x^2 - 3x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: One-Sided Limits and Rational Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate one-sided limits, especially when the denominator approaches zero.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"One-sided limit: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to 3^-\"}},{\"type\":\"text\",\"text\":\" means approaching $3"},{"type":"inlineMath","attrs":{"latex":" from the left.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Check for sign of numerator and denominator as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches $3"}},{"type":"text","text":".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Factor denominator: \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x^2 - 3x = x(x - 3)\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to 3^-\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x - 3\"}},{\"type\":\"text\",\"text\":\" approaches $0"},{"type":"inlineMath","attrs":{"latex":" from the negative side.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Analyze the sign of numerator and denominator near \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x = 3\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to 3^-} \\\\frac{x - 4}{x^2 - 3x} = -\\\\infty\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The denominator approaches $0"}},{"type":"text","text":" from the negative side, so the limit is negative infinity.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q12. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} e^{-2x} + \\\\frac{2}{x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits at Infinity\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate limits as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches infinity, especially with exponential and rational terms.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^{-2x}\"}},{\"type\":\"text\",\"text\":\": Exponential decay\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{2}{x}\"}},{\"type\":\"text\",\"text\":\": Rational function\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Analyze \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^{-2x}\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}},{\"type\":\"text\",\"text\":\" (goes to $0"},{"type":"inlineMath","attrs":{"latex":").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Analyze \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{2}{x}\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}},{\"type\":\"text\",\"text\":\" (goes to $0"}},{"type":"text","text":").\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Add the two limits together.\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} e^{-2x} + \\\\frac{2}{x} = 0\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Both terms approach "},{"type":"inlineMath","attrs":{"latex":"0\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" as \"}},{\"type\":\"text\",\"text\":\"x"}},{"type":"text","text":" goes to infinity.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q13. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} 3 \\\\tan^{-1} x + 2\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits at Infinity and Inverse Trigonometric Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate limits involving inverse tangent as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches infinity.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tan^{-1} x\"}},{\"type\":\"text\",\"text\":\" approaches \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{\\\\pi}{2}\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Recall that \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tan^{-1} x \\\\to \\\\frac{\\\\pi}{2}\"}},{\"type\":\"text\",\"text\":\" as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Multiply by "},{"type":"inlineMath","attrs":{"latex":"3\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" and add $2\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} 3 \\\\tan^{-1} x + 2 = \\\\frac{3\\\\pi}{2} + 2\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches infinity, \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\tan^{-1} x\"}},{\"type\":\"text\",\"text\":\" approaches \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{\\\\pi}{2}\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q14. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to -\\\\infty} \\\\frac{1}{2 + e^x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits at Infinity and Exponential Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate limits as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches negative infinity.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^x\"}},{\"type\":\"text\",\"text\":\" approaches $0\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" as \"}},{\"type\":\"text\",\"text\":\"x \\\\to -\\\\infty"}},{"type":"text","text":"\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to -\\\\infty\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^x \\\\to 0\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Substitute "},{"type":"inlineMath","attrs":{"latex":"0\"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\" for \"}},{\"type\":\"text\",\"text\":\"e^x"}},{"type":"text","text":" in the denominator.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Evaluate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\frac{1}{2}\"}},{\"type\":\"text\",\"text\":\".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to -\\\\infty} \\\\frac{1}{2 + e^x} = \\\\frac{1}{2}\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-\\\\infty\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^x\"}},{\"type\":\"text\",\"text\":\" becomes negligible.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q15. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} \\\\frac{1}{2 + e^x}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits at Infinity and Exponential Functions\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate limits as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches positive infinity.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^x\"}},{\"type\":\"text\",\"text\":\" approaches infinity as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^x \\\\to \\\\infty\"}},{\"type\":\"text\",\"text\":\".\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The denominator becomes very large.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The fraction approaches $0"},{"type":"inlineMath","attrs":{"latex":".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} \\\\frac{1}{2 + e^x} = 0\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"text\":\" approaches infinity, the denominator grows without bound.\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q16. Calculate \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} \\\\frac{\\\\cos x}{e^{3x}}\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\".\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Background\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"Topic: Limits at Infinity with Oscillating Numerator\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"This question tests your ability to evaluate limits where the numerator oscillates and the denominator grows rapidly.\"}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Key Terms and Formulas:\"}]},{\"type\":\"bulletList\",\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos x\"}},{\"type\":\"text\",\"text\":\" oscillates between \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"-1\"}},{\"type\":\"text\",\"text\":\" and $1"}},{"type":"text","text":"\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^{3x}\"}},{\"type\":\"text\",\"text\":\" grows rapidly as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Step-by-Step Guidance\"}]},{\"type\":\"orderedList\",\"attrs\":{\"start\":1,\"type\":null},\"content\":[{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"As \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x \\\\to \\\\infty\"}},{\"type\":\"text\",\"text\":\", \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"e^{3x}\"}},{\"type\":\"text\",\"text\":\" increases without bound.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\cos x\"}},{\"type\":\"text\",\"text\":\" remains bounded.\"}]}]},{\"type\":\"listItem\",\"content\":[{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The fraction approaches $0"},{"type":"inlineMath","attrs":{"latex":".\"}]}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"underline\"}],\"text\":\"Try solving on your own before revealing the answer!\"}]},{\"type\":\"collapsible\",\"content\":[{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":4},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Final Answer:\"}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"\\\\lim_{x \\\\to \\\\infty} \\\\frac{\\\\cos x}{e^{3x}} = 0\"}}]},{\"type\":\"paragraph\",\"attrs\":{\"textAlign\":null},\"content\":[{\"type\":\"text\",\"text\":\"The denominator dominates, so the limit is $0"}},{"type":"text","text":".\"}]}]},{\"type\":\"heading\",\"attrs\":{\"textAlign\":null,\"level\":3},\"content\":[{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\"Q17. Find the vertical asymptote by evaluating the function as \"},{\"type\":\"inlineMath\",\"attrs\":{\"latex\":\"x\"}},{\"type\":\"text\",\"marks\":[{\"type\":\"bold\"}],\"text\":\" approaches values of discontinuity: \"},{