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: Analyze left - hand limit

As $x$ approaches $- 5$ from the left ($x
ightarrow - 5^{-}$), observe the graph to find the value the function approaches.

Step3: Analyze right - hand limit

As $x$ approaches $- 5$ from the right ($x
ightarrow - 5^{+}$), observe the graph to find the value the function approaches.

Step4: Check function value

Find $f(-5)$ by looking at the $y$ - value of the graph at $x=-5$.

Step5: Compare values

If $\lim_{x
ightarrow - 5^{-}}f(x)=\lim_{x
ightarrow - 5^{+}}f(x)=f(-5)$, the function is continuous at $x = - 5$; otherwise, it is not.

Since the graph is not provided completely in a way that we can precisely read off the limits and function - value at $x=-5$, assume from a general understanding of continuity and graphs. If there is a break, jump or hole at $x = - 5$ in the graph, the function is discontinuous. Without seeing the exact behavior at $x=-5$ precisely, we can't give a definite numerical answer. But if we assume there is a break in the graph at $x=-5$ (for example, if the left - hand and right - hand limits are different or if the function value at $x=-5$ is not equal to the limit), the function is discontinuous.

Answer:

The function is discontinuous at $x=-5$ if the left - hand limit, right - hand limit and the function value at $x = - 5$ are not equal. Without a more detailed graph, we can't give a definite "yes" or "no" answer, but based on the general concept of continuity and the appearance of a possible break in the graph, it is likely discontinuous.