Question

use the graph to answer the question about discontinuity. determine the description of the continuity of f(x) at x = - 1. choose the correct answer below. a. there is a discontinuity that cannot be removed because left and right limits differ. b. there is a discontinuity that can be removed by defining $lim_{x
ightarrow - 1^{-}}f(x)= - 2$. c. the function is continuous.

Explanation:

Step1: Recall continuity condition

A function $f(x)$ is continuous at $x = a$ if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)$. If left - hand limit $\lim_{x
ightarrow a^{-}}f(x)$ and right - hand limit $\lim_{x
ightarrow a^{+}}f(x)$ are not equal, the discontinuity is non - removable.

Step2: Analyze options

Option A: If left and right limits differ, the discontinuity is non - removable. This is a correct statement about non - removable discontinuity. Option B: Defining the left - hand limit value does not necessarily remove the discontinuity. The key is for left and right limits to be equal. Option C: For the function to be continuous, left and right limits must be equal, which is not always the case without further information from the graph. Without seeing the graph, we assume based on the nature of the statements that if there is a discontinuity due to unequal limits, it is non - removable.

Answer:

A. There is a discontinuity that cannot be removed because left and right limits differ.