(b) $\lim _{x \rightarrow \infty} \tan ^{-1}\left(e^{x}\right)$

Step 1 ：First, we need to understand the meaning of the question. The question is asking for the limit as x approaches infinity of the inverse tangent of e to the power of x.

Step 2 ：We know that the inverse tangent function, or arctan, is a function that gives the angle whose tangent is any given number. It is the inverse function of the tangent function, which gives the ratio of the opposite side to the adjacent side of a right triangle.

Step 3 ：Since the limit is as x approaches infinity, we are essentially looking at the inverse tangent of an infinitely large number.

Step 4 ：We know that as the input to the tangent function gets larger and larger, the output of the function approaches \(\frac{\pi}{2}\), but never quite reaches it. This is because the tangent function has vertical asymptotes at \(\frac{\pi}{2}\) and \(-\frac{\pi}{2}\).

Step 5 ：Therefore, as the input to the inverse tangent function gets larger and larger, the output of the function approaches \(\frac{\pi}{2}\), but never quite reaches it.

Step 6 ：So, \(\lim _{x \rightarrow \infty} \tan ^{-1}\left(e^{x}\right) = \frac{\pi}{2}\).

Step 7 ：Finally, we check our answer. As x approaches infinity, e to the power of x also approaches infinity. The inverse tangent of an infinitely large number is \(\frac{\pi}{2}\), so our answer is correct.

Step 8 ：\(\boxed{\frac{\pi}{2}}\) is the final answer.