Let's analyze the given graph to determine the following limits:

a) $\lim_{{x \to -2^-}} f(x)$

This is asking for the limit of $f(x)$ as $x$ approaches $-2$ from the left-hand side (i.e., $x \to -2^-$).

From the graph:

• As $x$ approaches $-2$ from the left, the function approaches the value $1$.

Thus, $\lim_{{x \to -2^-}} f(x) = 1$.

b) $\lim_{{x \to -2^+}} f(x)$

This is asking for the limit of $f(x)$ as $x$ approaches $-2$ from the right-hand side (i.e., $x \to -2^+$).

From the graph:

• As $x$ approaches $-2$ from the right, the function approaches the value $3$.

Thus, $\lim_{{x \to -2^+}} f(x) = 3$.

c) $\lim_{{x \to -2}} f(x)$

This is asking for the two-sided limit of $f(x)$ as $x$ approaches $-2$ from both sides.

Since the left-hand limit $\lim_{{x \to -2^-}} f(x) = 1$ and the right-hand limit $\lim_{{x \to -2^+}} f(x) = 3$, the two-sided limit does not exist because the left and right limits are not equal.

Thus, $\lim_{{x \to -2}} f(x)$ does not exist.

Would you like any more details or clarifications?

Related Questions:

1. What is a one-sided limit, and how does it differ from a two-sided limit?
2. How do discontinuities in a graph affect limits?
3. What happens if the left-hand and right-hand limits are equal at a point?
4. Can limits exist at points where the function is undefined?
5. How can limits help in finding the behavior of a function near discontinuities?

Tip:

When evaluating limits from a graph, always check the direction from which $x$ approaches and whether the function has any jumps or breaks at that point.