Let's solve the given problems based on the graph:

Analysis:

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

To evaluate the two-sided limit as $x \to 2$, we examine the graph on both sides of $x = 2$:

• As $x \to 2^-$, $f(x) \to 1$ (approaching from the left).
• As $x \to 2^+$, $f(x) \to 2$ (approaching from the right).

Since the left and right limits do not match, the two-sided limit does not exist.

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

From the graph, as $x \to 2^+$ (approaching from the right), $f(x) \to 2$.

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

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

At $x = -2$:

• From the left ($x \to -2^-$), $f(x) \to -1$.
• From the right ($x \to -2^+$), $f(x) \to -1$.

Since the left and right limits agree, the two-sided limit exists, and $\lim_{x \to -2} f(x) = -1$.

(d) $f(-2)$

The value of $f(-2)$ is directly read from the graph. At $x = -2$, there is a filled point at $f(-2) = -1$.

Thus, $f(-2) = -1$.

Final Answers:

(a) $\lim_{x \to 2} f(x)$: Does not exist
(b) $\lim_{x \to 2^+} f(x) = 2$
(c) $\lim_{x \to -2} f(x) = -1$
(d) $f(-2) = -1$

Would you like a deeper explanation for any of these?

Related Questions:

1. How do you determine limits when the graph shows oscillations near a point?
2. Why is continuity not required for a limit to exist at a point?
3. What is the difference between one-sided and two-sided limits in terms of the graph?
4. How does the value of a function at a point relate to its limit?
5. Can a function have a limit at a point where it is not defined?

Tip: When solving graph-based limit problems, always check both sides of the target point for consistency.