Answer
The limit does not exist at $c=4$, but does exist at $c=2$ and equals to $+\infty$.
vertical asymptotes: $x = 2$ and $x = 4$
Work Step by Step
From the given figure, we find the limits
\begin{align*}
\lim _{x \rightarrow 2^{-}} f(x)&=+\infty \\
\lim _{x \rightarrow 2^{+}} f(x)&=+\infty \\
\lim _{x \rightarrow 4^{-}} f(x)&=-\infty\\
\lim _{x \rightarrow 4^{+}} f(x)&= 10
\end{align*}
For the overall limit to exist, the left and right limits must match. Thus, the limit does not exist at $c=4$, but does exist at $c=2$ and equals to $+\infty$.
The vertical asymptotes are the vertical lines at $c = 2$ and $c = 4$, where the function decreases or increases without bound.
