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

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. The height of a baseball is given by "},{"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 over "},{"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 in feet)"}]}]},{"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":"Plug "},{"type":"inlineMath","attrs":{"latex":"t = 1"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"s(t)"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"s(1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plug "},{"type":"inlineMath","attrs":{"latex":"t = 1.5"}},{"type":"text","text":" into "},{"type":"inlineMath","attrs":{"latex":"s(t)"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"s(1.5)"}},{"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":"\\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 the numerator and denominator separately, but stop before dividing."}]}]}]},{"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 = 50"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"s(1.5) = -16(1.5)^2 + 60(1.5) + 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":"text","text":" ft/sec"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The baseball's average velocity over "},{"type":"inlineMath","attrs":{"latex":"[1, 1.5]"}},{"type":"text","text":" is 20 ft/sec."}]}]},{"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:"}]},{"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 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 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 (both sides)"}]}]}]},{"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":"Locate "},{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","text":" on the graph and find the value "},{"type":"inlineMath","attrs":{"latex":"f(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 1 from the left ("},{"type":"inlineMath","attrs":{"latex":"x < 1"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Observe the behavior as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches 1 from the right ("},{"type":"inlineMath","attrs":{"latex":"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 the overall limit 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) = 5"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to 1^-} f(x) = 5"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c. "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to 1^+} f(x) = 5"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"d. "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to 1} f(x) = 5"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"All values are 5, indicating the function is continuous at "},{"type":"inlineMath","attrs":{"latex":"x = 1"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Use the graph to evaluate the function on "},{"type":"inlineMath","attrs":{"latex":"[-1, 4]"}},{"type":"text","marks":[{"type":"bold"}],"text":" and determine where it 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:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Continuous: No breaks, jumps, or holes in the graph."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Discontinuous: The function has a break, jump, or hole at a specific "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" value."}]}]}]},{"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 from "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":" for any jumps, holes, or breaks."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values where the function is not connected or has a gap."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List those "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values as points of discontinuity."}]}]}]},{"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 is discontinuous at "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"These are the points where the graph has breaks or jumps."}]}]},{"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 the point."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Limit: The value a function approaches as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" gets close to a specific point."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Indeterminate form: Both numerator and denominator approach 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 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 very close to 0."}]}]}]},{"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 2 as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches 0."}]}]},{"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 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 trigonometric 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":"text","text":" (double angle identity)"}]}]},{"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":"\\sin 2x"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"\\sin x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\cos x"}},{"type":"text","text":" using the identity."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute into the limit: "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to 0} \\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":" in numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the limit for "},{"type":"inlineMath","attrs":{"latex":"2 \\cos x"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"x \\to 0"}},{"type":"text","text":", but stop before evaluating."}]}]}]},{"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 \\lim_{x \\to 0} \\cos x = 2 \\times 1 = 2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Using the identity and limit properties, the answer is 2."}]}]},{"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 points near the target value."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Limit: The value a function approaches as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" gets close to a specific point."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Table of values: Evaluating the function at points near "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{4}"}},{"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":"Choose values of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" close to "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\pi}{4}"}},{"type":"text","text":" from both sides (e.g., "},{"type":"inlineMath","attrs":{"latex":"0.99\\frac{\\pi}{4}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"1.01\\frac{\\pi}{4}"}},{"type":"text","text":")."}]}]},{"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 the quotient for each value and observe the trend as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\pi}{4}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Round your results to four decimal places and look for a pattern, but stop before stating the limit."}]}]}]},{"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} = 1.4142"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The table shows the function approaches approximately 1.4142 as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" gets close 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: Limits and Trigonometric Identities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use trigonometric identities to simplify and 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 = \\cos^2 x - \\sin^2 x"}},{"type":"text","text":" (double angle identity)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factoring and simplifying expressions"}]}]}]},{"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":"\\cos 2x"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\cos^2 x - \\sin^2 x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor the numerator as "},{"type":"inlineMath","attrs":{"latex":"(\\cos x - \\sin x)(\\cos x + \\sin x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the limit: "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to \\frac{\\pi}{4}} \\frac{(\\cos x - \\sin x)(\\cos x + \\sin x)}{\\cos x - \\sin x}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify by canceling "},{"type":"inlineMath","attrs":{"latex":"\\cos x - \\sin x"}},{"type":"text","text":" and prepare to evaluate "},{"type":"inlineMath","attrs":{"latex":"\\cos x + \\sin x"}},{"type":"text","text":" at "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{\\pi}{4}"}},{"type":"text","text":", but stop before plugging in."}]}]}]},{"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}} \\cos x + \\sin x = \\frac{\\sqrt{2}}{2} + \\frac{\\sqrt{2}}{2} = \\sqrt{2}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The limit is "},{"type":"inlineMath","attrs":{"latex":"\\sqrt{2}"}},{"type":"text","text":", which is approximately 1.4142."}]}]},{"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: Limits and Difference Quotients"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate a limit involving radicals, often used in derivative definitions."}]},{"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":"Rationalizing the numerator"}]}]}]},{"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":"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":"inlineMath","attrs":{"latex":"(\\sqrt{5x + h} - \\sqrt{5x})(\\sqrt{5x + h} + \\sqrt{5x}) = (5x + h) - 5x = h"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rewrite the limit as "},{"type":"inlineMath","attrs":{"latex":"\\lim_{h \\to 0} \\frac{h}{h(\\sqrt{5x + h} + \\sqrt{5x})}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify and prepare to evaluate as "},{"type":"inlineMath","attrs":{"latex":"h \\to 0"}},{"type":"text","text":", but stop before plugging in."}]}]}]},{"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{1}{\\sqrt{5x + h} + \\sqrt{5x}} = \\frac{1}{2\\sqrt{5x}}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"As "},{"type":"inlineMath","attrs":{"latex":"h"}},{"type":"text","text":" approaches 0, the denominator becomes "},{"type":"inlineMath","attrs":{"latex":"2\\sqrt{5x}"}},{"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 Polynomial Division"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to simplify a polynomial expression and evaluate a limit."}]},{"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":"Factorization: "},{"type":"inlineMath","attrs":{"latex":"p^5 - 1 = (p - 1)(p^4 + p^3 + p^2 + p + 1)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Limit as "},{"type":"inlineMath","attrs":{"latex":"p \\to 1"}}]}]}]},{"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 "},{"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":"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":"Set up the limit: "},{"type":"inlineMath","attrs":{"latex":"\\lim_{p \\to 1} (p^4 + p^3 + p^2 + p + 1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Prepare to substitute "},{"type":"inlineMath","attrs":{"latex":"p = 1"}},{"type":"text","text":", but stop before calculating the sum."}]}]}]},{"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} (p^4 + p^3 + p^2 + p + 1) = 1 + 1 + 1 + 1 + 1 = 5"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The answer is 5."}]}]},{"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 Polynomial Factorization"}]},{"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":"Factorization: "},{"type":"inlineMath","attrs":{"latex":"x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Limit as "},{"type":"inlineMath","attrs":{"latex":"x \\to 3"}}]}]}]},{"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 "},{"type":"inlineMath","attrs":{"latex":"x^4 - 81"}},{"type":"text","text":" completely."}]}]},{"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":"Set up the limit: "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to 3} (x + 3)(x^2 + 9)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Prepare to substitute "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","text":", but stop before multiplying out."}]}]}]},{"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} (x + 3)(x^2 + 9) = (3 + 3)(3^2 + 9) = 6 \\times 18 = 108"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The answer is 108."}]}]},{"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 Behavior Near Discontinuity"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze one-sided limits and recognize infinite behavior."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"One-sided limit: Approaching from the left ("},{"type":"inlineMath","attrs":{"latex":"x \\to 3^-"}},{"type":"text","text":")"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Discontinuity: Denominator approaches zero"}]}]}]},{"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 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"}},{"type":"text","text":" approaches 3 from the left, "},{"type":"inlineMath","attrs":{"latex":"x - 3"}},{"type":"text","text":" is negative and close to zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Analyze the sign of numerator and denominator as "},{"type":"inlineMath","attrs":{"latex":"x \\to 3^-"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the limit, but stop before stating the infinite result."}]}]}]},{"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 limit approaches "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"x \\to 3^-"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The denominator approaches zero from the negative side, causing the fraction to diverge negatively."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. 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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:"}]},{"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 3 and add 2 to set up the limit."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Prepare to substitute, but stop before calculating the final value."}]}]}]},{"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 = 3 \\times \\frac{\\pi}{2} + 2 = \\frac{3\\pi}{2} + 2"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The answer is "},{"type":"inlineMath","attrs":{"latex":"\\frac{3\\pi}{2} + 2"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. 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Find the vertical asymptote(s) for "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x^2 - 5x + 6}{x^2 - 2x}"}},{"type":"text","marks":[{"type":"bold"}],"text":" by evaluating the function as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","marks":[{"type":"bold"}],"text":" approaches values of discontinuity."}]},{"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: Asymptotes and Discontinuities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find vertical asymptotes by analyzing where the denominator is zero and the function is undefined."}]},{"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":"Vertical asymptote: "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values where denominator is zero and numerator is not zero."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor numerator and denominator."}]}]}]},{"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 numerator: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 5x + 6 = (x - 3)(x - 2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor denominator: "},{"type":"inlineMath","attrs":{"latex":"x^2 - 2x = x(x - 2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" values where denominator is zero: "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if numerator is also zero at those points to determine if it's a vertical asymptote or removable discontinuity, but stop before stating which are asymptotes."}]}]}]},{"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":"Vertical asymptote at "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"At "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":", both numerator and denominator are zero, so it's a removable discontinuity, not a vertical asymptote."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q18. Analyze "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to \\infty} f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"\\lim_{x \\to -\\infty} f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x^2 - 5x + 6}{x^2 - 2x}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and determine if "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" has a slant asymptote."}]},{"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: End Behavior and Asymptotes"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze limits at infinity and determine horizontal or slant asymptotes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal asymptote: If degree of numerator equals denominator, limit is ratio of leading coefficients."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slant (oblique) asymptote: If degree of numerator is one more than denominator."}]}]}]},{"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":"Compare degrees of numerator and denominator (both degree 2)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the ratio of leading coefficients as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x \\to -\\infty"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the limits, but stop before stating the value or asymptote type."}]}]}]},{"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":"Both limits are 1, so "},{"type":"inlineMath","attrs":{"latex":"y = 1"}},{"type":"text","text":" is a horizontal asymptote."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"No slant asymptote since degrees are equal."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q19. Is "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" continuous at "},{"type":"inlineMath","attrs":{"latex":"x = 5"}},{"type":"text","marks":[{"type":"bold"}],"text":"? 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