Evaluate the following limit. If the limit does not exist, select 'DNE'.
$$\underset{x\to \infty }{lim}arctan\left(e^x\right)$$

• A. $\frac{\pi }{2}$
• B. $0$
• C. $1$
• D. DNE

Answer: (A)

Verified step by step guidance

1. Understand the problem: We are tasked with evaluating the limit of \( \arctan(e^x) \) as \( x \to \infty \). The goal is to determine the behavior of the function \( \arctan(e^x) \) as \( x \) becomes very large.
2. Recall the properties of the \( \arctan \) function: The \( \arctan(x) \) function is defined for all real numbers and has horizontal asymptotes at \( \frac{\pi}{2} \) as \( x \to \infty \) and \( -\frac{\pi}{2} \) as \( x \to -\infty \).
3. Analyze the behavior of \( e^x \) as \( x \to \infty \): The exponential function \( e^x \) grows without bound as \( x \to \infty \). This means \( e^x \to \infty \).
4. Substitute \( e^x \to \infty \) into \( \arctan(e^x) \): Since \( \arctan(x) \to \frac{\pi}{2} \) as \( x \to \infty \), it follows that \( \arctan(e^x) \to \frac{\pi}{2} \) as \( x \to \infty \).
5. Conclude the solution: The limit of \( \arctan(e^x) \) as \( x \to \infty \) exists and is equal to \( \frac{\pi}{2} \).