Question

use the definition of continuity to determine whether the function $f(x)$ graphed below is continuous at $x = - 5$.

Explanation:

Step1: Recall continuity definition

A function $f(x)$ is continuous at $x = a$ if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)$.

Step2: Find left - hand limit

As $x$ approaches $- 5$ from the left ($x
ightarrow - 5^{-}$), by looking at the graph, the function values approach a non - defined value (there is a hole). Let's assume we can estimate the value the function is approaching. The left - hand limit $\lim_{x
ightarrow - 5^{-}}f(x)$ exists and is some value.

Step3: Find right - hand limit

As $x$ approaches $-5$ from the right ($x
ightarrow - 5^{+}$), from the graph, the function values approach the same non - defined value as in the left - hand limit. So $\lim_{x
ightarrow - 5^{+}}f(x)=\lim_{x
ightarrow - 5^{-}}f(x)$.

Step4: Check function value

The function is defined at $x=-5$ (there is a filled - in dot). But the value of the function at $x = - 5$ is not equal to the limit of the function as $x$ approaches $-5$. Since $\lim_{x
ightarrow - 5}f(x)
eq f(-5)$ (where $\lim_{x
ightarrow - 5}f(x)$ is the common value of the left - hand and right - hand limits), the function is not continuous at $x=-5$.

Answer:

The function $f(x)$ is not continuous at $x = - 5$.