https://file.aiquickdraw.com/mathbot/file/1729798205121kav8y2sw.jpg

The expression in the image is:

\[
\lim_{{x \to \infty}} \arctan(e^x)
\]

### Solution:

1. As \( x \to \infty \), \( e^x \) grows very rapidly and tends towards infinity.
2. The arctangent function, \( \arctan(y) \), has a horizontal asymptote at \( \frac{\pi}{2} \) as \( y \to \infty \).
3. Therefore:

\[
\lim_{{x \to \infty}} \arctan(e^x) = \frac{\pi}{2}
\]

So the limit evaluates to \( \frac{\pi}{2} \).

Would you like further details or explanations?

Here are 5 related questions for practice:
1. What is \( \lim_{{x \to -\infty}} \arctan(e^x) \)?
2. How does \( \arctan(x) \) behave as \( x \to 0 \)?
3. What is the derivative of \( \arctan(e^x) \) with respect to \( x \)?
4. What happens to \( \lim_{{x \to 0}} \arctan(e^x) \)?
5. How does \( \arctan(x) \) compare to \( \arcsin(x) \) for large \( x \)?

**Tip**: When solving limits involving exponential or trigonometric functions, always keep asymptotic behavior in mind to simplify the analysis.

Explore how the limit of arctan(e^x) behaves as x approaches infinity. The exponential function grows rapidly, and the arctan function approaches a horizontal asymptote at π/2. Learn the step-by-step solution to this limit problem.

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