Question

use the graph to answer the question about discontinuity. determine the description of the continuity of f(x) at x=3. choose the correct answer below. a. there is a discontinuity that can be removed by defining f(3)=0. b. there is a discontinuity that cannot be removed. c. the function is continuous.

Explanation:

To determine continuity at \( x = 3 \), we check the limit as \( x \) approaches 3 and the function's value at \( x = 3 \). A removable discontinuity occurs when the limit exists (left - hand limit = right - hand limit) but the function value at the point is either undefined or different from the limit. From the graph, as \( x \) approaches 3, the left - hand limit and right - hand limit of \( f(x) \) are equal (the y - value that the graph approaches from both sides is 0). The function has a hole at \( x = 3 \), meaning \( f(3) \) is not equal to this limit (or is undefined at that point in the way of the hole). So, we can remove the discontinuity by defining \( f(3) \) to be equal to this limit (which is 0). Option B is incorrect because the discontinuity is removable. Option C is incorrect as there is a hole (discontinuity) at \( x = 3 \).

Answer:

A. There is a discontinuity that can be removed by defining \( f(3)=0 \)