To determine if the function $h(x)$ is continuous at $x = -2$, we need to check the following:

1. The function value at $x = -2$.
2. The left-hand limit as $x \to -2^-$.
3. The right-hand limit as $x \to -2^+$.

Step-by-Step Solution:

1. Compute the function value at $x = -2$:

Given that for $x \leq -2$, $h(x) = e^{x+2}$, we find:

$h(-2) = e^{-2 + 2} = e^0 = 1$

2. Compute the left-hand limit as $x \to -2^-$:

The left-hand limit corresponds to the part of the function where $x \leq -2$:

$\lim_{x \to -2^-} h(x) = \lim_{x \to -2^-} e^{x+2} = e^0 = 1$

3. Compute the right-hand limit as $x \to -2^+$:

The right-hand limit corresponds to the part of the function where $x > -2$:

$\lim_{x \to -2^+} h(x) = \lim_{x \to -2^+} (x+2)^e = (0)^e = 0$

Conclusion:

• The left-hand limit as $x \to -2^-$ is $1$.
• The right-hand limit as $x \to -2^+$ is $0$.
• The function value at $x = -2$ is $1$.

Since the left-hand limit does not equal the right-hand limit, the function $h(x)$ is not continuous at $x = -2$.

Answer:

B) No

Would you like more details, or do you have any questions? Here are some related questions to explore:

1. What is the definition of continuity at a point?
2. How do we check if a piecewise function is differentiable at a point?
3. What are the common types of discontinuities in functions?
4. How can the concept of limits help in understanding the behavior of functions at points of discontinuity?
5. How do we apply the concept of continuity in real-world problems?

Tip: Always check both the left-hand and right-hand limits to determine if a function is continuous at a point.