Question

use the graph of the function f shown to the right to estimate the indicated function values and limits. complete parts (a) through (e) below. ... (e) is f continuous at x = 2? explain. a. the function is continuous. b. the function is discontinuous at x=2 since $limlimits_{x\to 2} f(x)$ does not exist. c. the function is discontinuous at x=2 since f(2) does not exist. d. the function is discontinuous at x=2 since $limlimits_{x\to 2} f(x)
eq f(2)$.

Explanation:

To determine continuity at \( x = 2 \), we check three conditions: 1) \( f(2) \) exists, 2) \( \lim_{x \to 2} f(x) \) exists, and 3) \( \lim_{x \to 2} f(x)=f(2) \). From the graph (and typical discontinuity cases here), if the limit as \( x \to 2 \) does not equal \( f(2) \), the function is discontinuous. Option D states the function is discontinuous at \( x = 2 \) since \( \lim_{x \to 2} f(x)
eq f(2) \), which matches the discontinuity condition (third condition fails). Option B is incorrect if the limit exists but doesn't match \( f(2) \); option C is incorrect if \( f(2) \) exists. Option A is wrong as the function isn't continuous.

Answer:

D. The function is discontinuous at \( x = 2 \) since \( \lim\limits_{x \to 2} f(x)
eq f(2) \)