Calculus (3rd Edition)

The limit does not exist at $c=1,2$, but does exist at $c=4$ and equals to $2$.
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 $2$.