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AP Calculus BC:1.3 Estimating Limit  Values from Graphs- Exam Style questions with Answer- FRQ

Question

The function f is defined on the interval [-5, 5]. The graph of f is shown below.
[Insert a graph here. It should be a piecewise function with some discontinuities, perhaps a hole, a jump, and an asymptote. Make sure some values are clearly marked. For example, you could have f(-3) = 2, a hole at (-1, 4), a jump discontinuity at x=1 with f(1) = 3 and the limit from the right being 5, and a vertical asymptote at x=4.]
Use the graph to answer the following questions.
(a) Estimate \(\lim_{x \to -3}\) f(x).
(b) Estimate \(\lim_{x \to -1}\) f(x).
(c) Estimate \(\lim_{x \to 1^-}\) f(x).
(d) Estimate \(\lim_{x \to 1^+}\) f(x).
(e) State whether f is continuous at x = 1. Justify your answer.
(f) For what value(s) of x does f have a discontinuity? Classify each discontinuity as removable (hole), jump, or vertical asymptote.

▶️Answer/Explanation

ans:

(a) \(\lim_{x \to -3}\) f(x) = 2. (The function is continuous at x = -3, so the limit is simply the function value.)
(b) \(\lim_{x \to -1}\) f(x) = 4. (The function has a hole at x = -1. The limit exists and is the y-value of the hole, even though the function is not defined there.)
(c) \(\lim_{x \to 1^-}\) f(x) = 3. (This is the limit as x approaches 1 from the left.)
(d) \(\lim_{x \to 1^+}\) f(x) = 5. (This is the limit as x approaches 1 from the right.)
(e) f is not continuous at x = 1. The limit from the left and the limit from the right are not equal, so the limit does not exist. For a function to be continuous at a point, the limit must exist and be equal to the function value.
(f) f has a removable discontinuity (hole) at x = -1, a jump discontinuity at x = 1, and a vertical asymptote at x = 4.

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