https://file.aiquickdraw.com/mathbot/file/1726859973644g6xkwh2w.jpg

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.

The graphs of the functions g(x) and h(x) are shown. Determine the following limits: (a) lim (x→-2) h(x), (b) lim (x→1) g(h(x)), (c) lim (x→-2) h(g(x)).

Explore how to determine limits from the graphs of two functions, g(x) and h(x). The problem covers concepts like limits, continuity, and evaluating composite function limits. This solution is aimed at students in Grades 11-12.

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