{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. Evaluate the following limits for the given graph. Give your answer as a finite number, "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","marks":[{"type":"bold"}],"text":", "},{"type":"inlineMath","attrs":{"latex":"\\infty"}},{"type":"text","marks":[{"type":"bold"}],"text":", or DNE."}]},{"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 from Graphs"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to read and interpret limits from a graph, including one-sided and two-sided limits, as well as function values at specific points."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"One-sided limit:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a^-} f(x)"}},{"type":"text","text":" (from the left), "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a^+} f(x)"}},{"type":"text","text":" (from the right)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Two-sided limit:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} f(x)"}},{"type":"text","text":" (exists only if both one-sided limits are equal)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"DNE:"},{"type":"text","text":" Does Not Exist (if the left and right limits are not equal or the function diverges)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Function value:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(a)"}},{"type":"text","text":" is the value of the function at "},{"type":"inlineMath","attrs":{"latex":"x = a"}},{"type":"text","text":" (may not equal the limit)"}]}]}]},{"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":"For each limit, identify whether it is a left-hand, right-hand, or two-sided limit. For example, "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 3^-} f(x)"}},{"type":"text","text":" is the limit as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches 3 from the left."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Examine the graph near the point of interest. Observe the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-values as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches the specified value from the correct direction."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"If the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-values approach a finite number, that is the limit. If they increase or decrease without bound, the limit is "},{"type":"inlineMath","attrs":{"latex":"\\infty"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","text":". If the left and right limits are not equal, the two-sided limit does not exist (DNE)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For function values like "},{"type":"inlineMath","attrs":{"latex":"f(-3)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"f(-2)"}},{"type":"text","text":", or "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":", look for a solid dot (indicating the function is defined there) and read the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-value at that "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For limits as "},{"type":"inlineMath","attrs":{"latex":"x \\to -\\infty"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":", observe the end behavior of the graph as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" moves far to the left or right."}]}]}]},{"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 Answers:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. $4$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. $4$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c. $4$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"d. "},{"type":"inlineMath","attrs":{"latex":"\\infty"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"e. "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f. DNE"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"g. $1$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"h. $0$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"i. DNE"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"j. $1$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"k. DNE"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"l. $0$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"m. $3$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"n. "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each answer is determined by carefully reading the graph at the specified points and directions."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. For the graph of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":", state the intervals on which the function is continuous. For each discontinuity, state which conditions required by the Continuity Test are not met. Classify each 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: Continuity and Types of Discontinuities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of where a function is continuous and how to classify discontinuities (removable, jump, infinite)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Concepts:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Continuity at a point "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","marks":[{"type":"bold"}],"text":":"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(a)"}},{"type":"text","text":" is defined, "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} f(x)"}},{"type":"text","text":" exists, and "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} f(x) = f(a)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Removable discontinuity:"},{"type":"text","text":" Limit exists, but "},{"type":"inlineMath","attrs":{"latex":"f(a)"}},{"type":"text","text":" is not defined or not equal to the limit."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Jump discontinuity:"},{"type":"text","text":" Left and right limits exist but are not equal."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Infinite discontinuity:"},{"type":"text","text":" Function approaches "},{"type":"inlineMath","attrs":{"latex":"\\infty"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","text":" near "},{"type":"inlineMath","attrs":{"latex":"a"}},{"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":"Identify all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-values where the graph is not continuous (breaks, holes, jumps, or vertical asymptotes)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each discontinuity, check which of the three conditions for continuity fail."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Classify each discontinuity as removable, jump, or infinite based on the graph's behavior."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List the intervals where the function is continuous (where the graph is unbroken)."}]}]}]},{"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":"Continuous on "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, -3) \\cup (-3, -2) \\cup (-2, 0) \\cup (0, \\infty)"}},{"type":"text","text":"."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Removable discontinuity at "},{"type":"inlineMath","attrs":{"latex":"x = -3"}},{"type":"text","text":" (limit exists, but "},{"type":"inlineMath","attrs":{"latex":"f(-3)"}},{"type":"text","text":" is not equal to the limit)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Infinite discontinuity at "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":" (function approaches "},{"type":"inlineMath","attrs":{"latex":"\\infty"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Jump discontinuity at "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" (left and right limits exist but are not equal)."}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Complete the statement for the definition of the limit "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} g(x) = L"}},{"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: Formal (Epsilon-Delta) Definition of a Limit"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the precise definition of a limit using "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon"}},{"type":"text","text":" (epsilon) and "},{"type":"inlineMath","attrs":{"latex":"\\delta"}},{"type":"text","text":" (delta)."}]},{"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","marks":[{"type":"bold"}],"text":"Epsilon ("},{"type":"inlineMath","attrs":{"latex":"\\varepsilon"}},{"type":"text","marks":[{"type":"bold"}],"text":"):"},{"type":"text","text":" Any positive number representing how close "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" should be to "},{"type":"inlineMath","attrs":{"latex":"L"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Delta ("},{"type":"inlineMath","attrs":{"latex":"\\delta"}},{"type":"text","marks":[{"type":"bold"}],"text":"):"},{"type":"text","text":" A positive number representing how close "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" should be to "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":" (but not equal to $a$)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Formal statement:"},{"type":"text","text":" For every "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon > 0"}},{"type":"text","text":", there exists "},{"type":"inlineMath","attrs":{"latex":"\\delta > 0"}},{"type":"text","text":" such that for all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":", if "},{"type":"inlineMath","attrs":{"latex":"0 < |x - a| < \\delta"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"|g(x) - L| < \\varepsilon"}},{"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":"Recall the structure: For every "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon > 0"}},{"type":"text","text":", there exists "},{"type":"inlineMath","attrs":{"latex":"\\delta > 0"}},{"type":"text","text":" such that for all "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"..."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The first blank is the condition on "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" (how close $x$ is to "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The second blank is the resulting closeness of "},{"type":"inlineMath","attrs":{"latex":"g(x)"}},{"type":"text","text":" to "},{"type":"inlineMath","attrs":{"latex":"L"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use absolute value notation to express these conditions."}]}]}]},{"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":"|x - a| < \\delta"}},{"type":"text","text":" implies "},{"type":"inlineMath","attrs":{"latex":"|g(x) - L| < \\varepsilon"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is the standard epsilon-delta definition of a limit."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. Write a proof for each of the following using the limit definition."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 2} (3x - 5) = 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to \\infty} (3x^2 - 2x + 1) / (3x^2 + 4x + 9) = 1"}}]}]}]},{"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: Epsilon-Delta Proofs and Limits at Infinity"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the formal definition of a limit to prove a limit statement, both for finite and infinite 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","marks":[{"type":"bold"}],"text":"Epsilon-Delta proof:"},{"type":"text","text":" Show that for every "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon > 0"}},{"type":"text","text":", there exists "},{"type":"inlineMath","attrs":{"latex":"\\delta > 0"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"|x - a| < \\delta"}},{"type":"text","text":" implies "},{"type":"inlineMath","attrs":{"latex":"|f(x) - L| < \\varepsilon"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Limits at infinity:"},{"type":"text","text":" For every "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon > 0"}},{"type":"text","text":", there exists "},{"type":"inlineMath","attrs":{"latex":"M"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"x > M"}},{"type":"text","text":" implies "},{"type":"inlineMath","attrs":{"latex":"|f(x) - L| < \\varepsilon"}},{"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":"For part (a), start by expressing "},{"type":"inlineMath","attrs":{"latex":"|3x - 5 - 1|"}},{"type":"text","text":" and simplify."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"|3x - 6| = 3|x - 2|"}},{"type":"text","text":" and relate this to "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"\\delta"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For part (b), divide numerator and denominator by "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" to analyze the limit as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\infty"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express "},{"type":"inlineMath","attrs":{"latex":"\\left| \\frac{3x^2 - 2x + 1}{3x^2 + 4x + 9} - 1 \\right|"}},{"type":"text","text":" and simplify."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the inequality and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"\\varepsilon"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"M"}},{"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":"text","text":"a. Let "},{"type":"inlineMath","attrs":{"latex":"\\delta = \\varepsilon / 3"}},{"type":"text","text":". If "},{"type":"inlineMath","attrs":{"latex":"|x - 2| < \\delta"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"|3x - 5 - 1| = 3|x - 2| < \\varepsilon"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. For "},{"type":"inlineMath","attrs":{"latex":"x > M = (4 + 9/\\varepsilon)/3"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\left| \\frac{3x^2 - 2x + 1}{3x^2 + 4x + 9} - 1 \\right| < \\varepsilon"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each proof follows the structure of the epsilon-delta (or "},{"type":"inlineMath","attrs":{"latex":"M"}},{"type":"text","text":"-"},{"type":"inlineMath","attrs":{"latex":"\\varepsilon"}},{"type":"text","text":") definition."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Evaluate the following limits or state that the limit does not exist."}]},{"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: Evaluating Limits (Algebraic, Trigonometric, Piecewise, and at Infinity)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute limits using algebraic simplification, factoring, rationalization, and knowledge of trigonometric and piecewise functions."}]},{"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","marks":[{"type":"bold"}],"text":"Limits at infinity:"},{"type":"text","text":" Focus on the highest degree terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Factoring and simplification:"},{"type":"text","text":" Useful for indeterminate forms like "},{"type":"inlineMath","attrs":{"latex":"0/0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Trigonometric limits:"},{"type":"text","text":" Use known limits such as "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 0} \\frac{\\sin x}{x} = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Piecewise functions:"},{"type":"text","text":" Evaluate the limit from the correct side and use the appropriate piece."}]}]}]},{"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":"For each limit, identify if it is at a finite value, at infinity, or involves a piecewise function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For rational functions at infinity, divide numerator and denominator by the highest power of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For limits at a point, factor or rationalize if you get "},{"type":"inlineMath","attrs":{"latex":"0/0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For trigonometric limits, recall standard limits and identities."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For piecewise functions, check which piece applies as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches the specified 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 Answers:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. 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"},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{5}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"h. "},{"type":"inlineMath","attrs":{"latex":"-\\frac{8}{3}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"i. "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"j. $4$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"k. $0$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"l. "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{6}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"m. $7$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"n. $5$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"o. "},{"type":"inlineMath","attrs":{"latex":"-\\infty"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"p. DNE"}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each answer is found by applying the appropriate limit laws and simplifications."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Find the following limit using the Sandwich Theorem: "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 0} x^2 \\cos \\frac{1}{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: Sandwich (Squeeze) Theorem"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the Sandwich Theorem to evaluate a limit where the function is bounded between two simpler functions."}]},{"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","marks":[{"type":"bold"}],"text":"Sandwich Theorem:"},{"type":"text","text":" If "},{"type":"inlineMath","attrs":{"latex":"g(x) \\leq f(x) \\leq h(x)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} g(x) = \\lim\\limits_{x \\to a} h(x) = L"}},{"type":"text","text":", then "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to a} f(x) = L"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Bounded functions:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"|\\cos \\frac{1}{x}| \\leq 1"}},{"type":"text","text":" for all "},{"type":"inlineMath","attrs":{"latex":"x \\neq 0"}},{"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":"Note that "},{"type":"inlineMath","attrs":{"latex":"-1 \\leq \\cos \\frac{1}{x} \\leq 1"}},{"type":"text","text":" for all "},{"type":"inlineMath","attrs":{"latex":"x \\neq 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply all parts of the inequality by "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" (which is always non-negative)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This gives "},{"type":"inlineMath","attrs":{"latex":"-x^2 \\leq x^2 \\cos \\frac{1}{x} \\leq x^2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the limits of "},{"type":"inlineMath","attrs":{"latex":"-x^2"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"x \\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\\limits_{x \\to 0} x^2 \\cos \\frac{1}{x} = 0"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"By the Sandwich Theorem, since both bounding functions approach $0"},{"type":"inlineMath","attrs":{"latex":" as "}},{"type":"text","text":"x \\to 0"},{"type":"inlineMath","attrs":{"latex":", so does "}},{"type":"text","text":"x^2 \\cos \\frac{1}{x}$."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Find the value of "},{"type":"inlineMath","attrs":{"latex":"a"}},{"type":"text","marks":[{"type":"bold"}],"text":" so that the function "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\begin{cases} 3x^2 - 13, & x < 2 \\\\ ax, & x \\geq 2 \\end{cases}"}},{"type":"text","marks":[{"type":"bold"}],"text":" is 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 of Piecewise Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find a parameter that makes a piecewise function continuous at a point."}]},{"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","marks":[{"type":"bold"}],"text":"Continuity at "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","marks":[{"type":"bold"}],"text":":"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 2^-} f(x) = \\lim\\limits_{x \\to 2^+} f(x) = f(2)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set the two pieces equal at "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":" to solve for "},{"type":"inlineMath","attrs":{"latex":"a"}},{"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":"Compute "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 2^-} f(x)"}},{"type":"text","text":" using the first piece: "},{"type":"inlineMath","attrs":{"latex":"3x^2 - 13"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 2^+} f(x)"}},{"type":"text","text":" using the second piece: "},{"type":"inlineMath","attrs":{"latex":"ax"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set these two limits equal to ensure continuity at "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the resulting equation for "},{"type":"inlineMath","attrs":{"latex":"a"}},{"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":"a = -\\frac{7}{4}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This value ensures the left and right limits at "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":" are equal, making the function continuous there."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. Define "},{"type":"inlineMath","attrs":{"latex":"h(4)"}},{"type":"text","marks":[{"type":"bold"}],"text":" in a way that extends "},{"type":"inlineMath","attrs":{"latex":"h(x) = \\frac{x^2 + 9x - 4}{x - 4}"}},{"type":"text","marks":[{"type":"bold"}],"text":" to be continuous at "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"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: Removable Discontinuity and Continuity Extension"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to remove a discontinuity by defining the function value at a point to match the 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","marks":[{"type":"bold"}],"text":"Removable discontinuity:"},{"type":"text","text":" Occurs when the limit exists but the function is undefined or not equal to the limit at that point."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To make "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","text":" continuous at "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":", set "},{"type":"inlineMath","attrs":{"latex":"h(4)"}},{"type":"text","text":" equal to "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 4} h(x)"}},{"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 the numerator if possible to simplify "},{"type":"inlineMath","attrs":{"latex":"h(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 4} h(x)"}},{"type":"text","text":" by canceling the "},{"type":"inlineMath","attrs":{"latex":"(x - 4)"}},{"type":"text","text":" factor."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set "},{"type":"inlineMath","attrs":{"latex":"h(4)"}},{"type":"text","text":" equal to this limit 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":"h(4) = 7"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Defining "},{"type":"inlineMath","attrs":{"latex":"h(4)"}},{"type":"text","text":" as $7"},{"type":"inlineMath","attrs":{"latex":" removes the discontinuity and makes "}},{"type":"text","text":"h(x)"},{"type":"inlineMath","attrs":{"latex":" continuous at "}},{"type":"text","text":"x = 4$."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Use the concept of composite functions to explain why "},{"type":"inlineMath","attrs":{"latex":"g(x) = \\cos(x^2)"}},{"type":"text","marks":[{"type":"bold"}],"text":" is a continuous function."}]},{"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 of Composite Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how the composition of continuous functions is also continuous."}]},{"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","marks":[{"type":"bold"}],"text":"Continuous function:"},{"type":"text","text":" A function is continuous if its limit at every point equals its value at that point."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Composite function:"},{"type":"text","text":" If "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"g"}},{"type":"text","text":" are continuous, then "},{"type":"inlineMath","attrs":{"latex":"f(g(x))"}},{"type":"text","text":" is 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":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\cos(x)"}},{"type":"text","text":" is continuous everywhere."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" is continuous everywhere."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The composition "},{"type":"inlineMath","attrs":{"latex":"\\cos(x^2)"}},{"type":"text","text":" is continuous because both "},{"type":"inlineMath","attrs":{"latex":"\\cos(x)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" are continuous."}]}]}]},{"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":"Since "},{"type":"inlineMath","attrs":{"latex":"\\cos(x)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"x^2"}},{"type":"text","text":" are both continuous everywhere, their composition "},{"type":"inlineMath","attrs":{"latex":"g(x) = \\cos(x^2)"}},{"type":"text","text":" is also continuous everywhere."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. Use the Intermediate Value Theorem to prove that "},{"type":"inlineMath","attrs":{"latex":"f(x) = x^3 - 7x^2 + 9x - 6"}},{"type":"text","marks":[{"type":"bold"}],"text":" has a solution between "},{"type":"inlineMath","attrs":{"latex":"x = 3"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"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: Intermediate Value Theorem (IVT)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to apply the IVT to show that a function has a root in a given 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":"text","marks":[{"type":"bold"}],"text":"IVT:"},{"type":"text","text":" If "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","text":" is continuous on "},{"type":"inlineMath","attrs":{"latex":"[a, b]"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"f(a)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"f(b)"}},{"type":"text","text":" have opposite signs, then there is at least one "},{"type":"inlineMath","attrs":{"latex":"c"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"(a, b)"}},{"type":"text","text":" such that "},{"type":"inlineMath","attrs":{"latex":"f(c) = 0"}},{"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":"Verify that "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" is continuous on "},{"type":"inlineMath","attrs":{"latex":"[3, 4]"}},{"type":"text","text":" (it is a polynomial, so it is continuous everywhere)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"f(3)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"f(4)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if "},{"type":"inlineMath","attrs":{"latex":"f(3)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"f(4)"}},{"type":"text","text":" have opposite signs."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Conclude that by the IVT, there is a root between $3"},{"type":"inlineMath","attrs":{"latex":" and $4"}},{"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":"f(3) = -3"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"f(4) = 50"}},{"type":"text","text":"; since "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" is continuous and changes sign between $3"},{"type":"inlineMath","attrs":{"latex":" and $4"}},{"type":"text","text":", by the IVT there is a solution in "},{"type":"inlineMath","attrs":{"latex":"(3, 4)"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q11. Find all asymptotes for the function "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{x^2 - 2}{x^2 - 8}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and then sketch its graph."}]},{"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 of Rational Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find vertical and horizontal asymptotes for rational functions."}]},{"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","marks":[{"type":"bold"}],"text":"Vertical asymptote:"},{"type":"text","text":" Set denominator equal to zero and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Horizontal asymptote:"},{"type":"text","text":" Compare degrees of 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":"Set "},{"type":"inlineMath","attrs":{"latex":"x^2 - 8 = 0"}},{"type":"text","text":" to find vertical asymptotes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare degrees: both numerator and denominator are degree 2, so horizontal asymptote is at "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{1}{1} = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check for holes (removable discontinuities) by factoring numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the graph, showing asymptotes and general behavior."}]}]}]},{"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 asymptotes at "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 2"}},{"type":"text","text":"; horizontal asymptote at "},{"type":"inlineMath","attrs":{"latex":"y = 1"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The graph approaches these lines but never crosses the vertical asymptotes."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. Sketch the graph of the function "},{"type":"inlineMath","attrs":{"latex":"g(x) = \\begin{cases} x^2 + 5, & x < 0 \\\\ 2, & x = 0 \\\\ x^2 - 5, & x > 0 \\end{cases}"}},{"type":"text","marks":[{"type":"bold"}],"text":" and then find each of the following limits:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 0^-} g(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 0^+} g(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c. Does "},{"type":"inlineMath","attrs":{"latex":"\\lim\\limits_{x \\to 0} g(x)"}},{"type":"text","text":" exist? Why or why not?"}]}]}]},{"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 of Piecewise Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate one-sided and two-sided limits for a piecewise function and to determine if the overall limit exists."}]},{"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","marks":[{"type":"bold"}],"text":"One-sided limits:"},{"type":"text","text":" Use the appropriate piece for "},{"type":"inlineMath","attrs":{"latex":"x < 0"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"x > 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Two-sided limit:"},{"type":"text","text":" Exists only if both one-sided 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":"For "},{"type":"inlineMath","attrs":{"latex":"x \\to 0^-"}},{"type":"text","text":", use "},{"type":"inlineMath","attrs":{"latex":"g(x) = x^2 + 5"}},{"type":"text","text":" and evaluate the limit as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches $0$ from the left."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"x \\to 0^+"}},{"type":"text","text":", use "},{"type":"inlineMath","attrs":{"latex":"g(x) = x^2 - 5"}},{"type":"text","text":" and evaluate the limit as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" approaches $0$ from the right."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compare the two one-sided limits to determine if the two-sided limit exists."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check the value at "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":" for completeness."}]}]}]},{"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 Answers:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. $5$"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. "},{"type":"inlineMath","attrs":{"latex":"-5"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"c. No, the limit does not exist because the left and right limits are not equal."}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. Find the average rate of change of the function "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\sin 2x"}},{"type":"text","marks":[{"type":"bold"}],"text":" on the interval "},{"type":"inlineMath","attrs":{"latex":"\\left[ \\frac{\\pi}{6}, \\frac{\\pi}{2} \\right]"}},{"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: Average Rate of Change"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to compute the average rate of change of a function over a given 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":"text","marks":[{"type":"bold"}],"text":"Average rate of change:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\frac{f(b) - f(a)}{b - a}"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"a < b"}},{"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":"Identify "},{"type":"inlineMath","attrs":{"latex":"a = \\frac{\\pi}{6}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"b = \\frac{\\pi}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Compute "},{"type":"inlineMath","attrs":{"latex":"f(a) = \\sin(2a)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"f(b) = \\sin(2b)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract "},{"type":"inlineMath","attrs":{"latex":"f(a)"}},{"type":"text","text":" from "},{"type":"inlineMath","attrs":{"latex":"f(b)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide by "},{"type":"inlineMath","attrs":{"latex":"b - a"}},{"type":"text","text":" to find the average rate of change."}]}]}]},{"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":"Average rate of change is "},{"type":"inlineMath","attrs":{"latex":"\\frac{3\\sqrt{3} - 3}{2\\pi}"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This is found by evaluating "},{"type":"inlineMath","attrs":{"latex":"\\sin(2x)"}},{"type":"text","text":" at the endpoints and applying the formula."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. For the function "},{"type":"inlineMath","attrs":{"latex":"f(x) = x^2 - 2x + 3"}},{"type":"text","marks":[{"type":"bold"}],"text":":"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. Find the slope of the curve at the point "},{"type":"inlineMath","attrs":{"latex":"(-1, 6)"}},{"type":"text","text":" using a limit."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. Find an equation of the tangent line to the curve at "},{"type":"inlineMath","attrs":{"latex":"(-1, 6)"}},{"type":"text","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: Derivative as a Limit and Tangent Lines"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the definition of the derivative to find the slope at a point and to write the equation of the tangent line."}]},{"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","marks":[{"type":"bold"}],"text":"Derivative at "},{"type":"inlineMath","attrs":{"latex":"x = a"}},{"type":"text","marks":[{"type":"bold"}],"text":":"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f'(a) = \\lim\\limits_{h \\to 0} \\frac{f(a + h) - f(a)}{h}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Tangent line:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"y = m(x - a) + f(a)"}},{"type":"text","text":", where "},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":" is the slope at "},{"type":"inlineMath","attrs":{"latex":"x = a"}},{"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":"Set up the difference quotient for "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" at "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify "},{"type":"inlineMath","attrs":{"latex":"f(-1 + h) - f(-1)"}},{"type":"text","text":" and divide by "},{"type":"inlineMath","attrs":{"latex":"h"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Take the limit as "},{"type":"inlineMath","attrs":{"latex":"h \\to 0"}},{"type":"text","text":" to find the slope."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the point-slope form to write the equation of the tangent line at "},{"type":"inlineMath","attrs":{"latex":"(-1, 6)"}},{"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 Answers:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a. Slope at "},{"type":"inlineMath","attrs":{"latex":"(-1, 6)"}},{"type":"text","text":" is "},{"type":"inlineMath","attrs":{"latex":"-5"}},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"b. Equation of the tangent line: "},{"type":"inlineMath","attrs":{"latex":"y = -5(x + 1) + 6"}},{"type":"text","text":" or "},{"type"