Let's analyze the graph and solve the limits step by step.

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

Looking at the graph of $h(x)$, as $x \to -2$, the value of $h(x)$ approaches 2 from both sides (left and right).

Thus, $\lim_{x \to -2} h(x) = 2.$

(b) $\lim_{x \to 1} g(h(x))$

First, find $h(1)$ from the graph of $h(x)$. At $x = 1$, $h(1) = 0$.

Next, we need to evaluate $g(0)$. From the graph of $g(x)$, $g(0) = 1$.

Thus, $\lim_{x \to 1} g(h(x)) = g(0) = 1.$

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

First, find $g(-2)$ from the graph of $g(x)$. At $x = -2$, $g(-2) = 2$.

Now, we need to evaluate $h(2)$. From the graph of $h(x)$, as $x \to 2$, there is a discontinuity. The left-hand limit $h(2^-) = 2$, and the right-hand limit $h(2^+) = 0$. Since these two limits are not equal, the limit does not exist.

Thus, $\lim_{x \to -2} h(g(x)) = \text{DNE} \ (\text{Does Not Exist}).$

Would you like further explanation on any of these points or details on limit properties?

Here are five related questions for further practice:

1. What is $\lim_{x \to 1} h(x)$?
2. What is $\lim_{x \to -1} g(x)$?
3. Determine $\lim_{x \to 3} h(x)$, if it exists.
4. What is $g(h(-1))$?
5. What is $h(g(1))$?

Tip: When evaluating composite limits like $g(h(x))$, always check both the inner and outer functions step by step.