Estimating a two-sided limit from a graph with a hole

As $x$ approaches $-1$ from the left, the curve rises smoothly and approaches a height of $4$ on the vertical axis. Similarly, as $x$ approaches $-1$ from the right, the curve also approaches a height of $4$. So both one-sided limits appear to equal $4$.

Since the left-hand limit and the right-hand limit are both $4$, the two-sided limit exists and equals $4$. The fact that the graph has an open circle at $x=-1$ (meaning $f(-1)$ is undefined) has no effect on this conclusion, because the limit only depends on values of $f(x)$ near $x=-1$, not at $x=-1$ itself.

No, it does not change the limit at all. The limit only depends on the values of the function near $x = -1$, not on the value at $x = -1$ itself. Even though $f(-1)$ is undefined, the curve still approaches the height of four from both sides. So the limit is four regardless of the hole.