$$ \lim_{h\to0}\text{ }\frac{1}{h}\int_2^{2+h}\!e^{-x^2}\,\mathrm{d}x $$

To evaluate the limit

\[ \lim_{h \to 0} \frac{1}{h} \int_2^{2+h} e^{-x^2} \, \mathrm{d}x, \]

we can recognize that this limit represents the derivative of a function defined by an integral. Specifically, let's define a function $ F(h) $ as follows:

\[ F(h) = \int_2^{2+h} e^{-x^2} \, \mathrm{d}x. \]

Then, the given limit is

\[ \lim_{h \to 0} \frac{F(h)}{h}. \]

This expression is the definition of the derivative of $ F(h) $ at $ h = 0 $. In other words, we have

\[ \lim_{h \to 0} \frac{F(h) - F(0)}{h}. \]

Since $ F(0) $ is

\[ F(0) = \int_2^2 e^{-x^2} \, \mathrm{d}x = 0, \]

the limit becomes

\[ \lim_{h \to 0} \frac{F(h)}{h} = F'(0). \]

To find $ F'(h) $, we use the Fundamental Theorem of Calculus and Leibniz's rule for differentiation under the integral sign:

\[ F'(h) = \frac{d}{dh} \left( \int_2^{2+h} e^{-x^2} \, \mathrm{d}x \right). \]

By Leibniz's rule, this derivative is

\[ F'(h) = e^{-(2+h)^2} \cdot \frac{d}{dh} (2 + h) - e^{-2^2} \cdot \frac{d}{dh} (2) + \int_2^{2+h} \frac{\partial}{\partial h} (e^{-x^2}) \, \mathrm{d}x. \]

Since $ e^{-2^2} \cdot \frac{d}{dh} (2) = 0 $ and $ \frac{\partial}{\partial h} (e^{-x^2}) = 0 $, the expression simplifies to

\[ F'(h) = e^{-(2+h)^2}. \]

Evaluating this at $ h = 0 $, we get

\[ F'(0) = e^{-2^2} = e^{-4}. \]

Therefore, the value of the limit is

\[ \boxed{e^{-4}}. \]

Would you like more details or have any questions?

Here are some related questions:

1. What is the Fundamental Theorem of Calculus?
2. How does Leibniz's rule apply to differentiation under the integral sign?
3. Can you explain the concept of a derivative in the context of a limit?
4. How do you evaluate definite integrals with variable limits?
5. What is the difference between $ F'(x) $ and $ \frac{d}{dx} F(x) $?
6. How do you interpret $ e^{-x^2} $ graphically?
7. What are some real-world applications of evaluating such limits?
8. How does this problem illustrate the use of the chain rule in calculus?

**Tip:** When dealing with limits involving integrals, recognizing the connection to the derivative of an integral function can simplify the problem significantly.

Explore how to evaluate the limit involving the integral of e^(-x^2) with detailed step-by-step explanations. This problem covers fundamental concepts in calculus, including limits, derivatives, and the application of Leibniz's rule. Suitable for advanced high school students and undergraduates.

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