Answer
The limit does not exist at $c=1,2$, but does exist at $c=4$ and equals to $2$.
Work Step by Step
From the given figure, we find the limits:
\begin{align*}
\lim _{x \rightarrow 1^{-}} f(x)&=3\\
\lim _{x \rightarrow 1^{+}} f(x)&=1\\
\lim _{x \rightarrow 2^{-}} f(x)&=2\\
\lim _{x \rightarrow 2^{+}} f(x)&=1\\
\lim _{x \rightarrow 4^{-}} f(x)&=2\\
\lim _{x \rightarrow 4^{+}} f(x)&=2
\end{align*}
For the overall limit to exist, the left and right limits must match. Thus, the limit does not exist at $c=1,2$, but does exist at $c=4$ and equals to $2$.
