Q. 22 - Calculus

Question

Find the limit by hand.

$\underset{x\to -\infty }{\lim }{e}^x {\tan }^{-1}x$

Step-by-Step Solution

Answer

On solving the limit, we get, $0$.

Step 1. Given information.

Consider the given question,

$\underset{x\to -\infty }{\lim }{e}^x {\tan }^{-1}x$

Step 2. Solve the given limit.

Using the product rule,

$$\begin{gathered} =\underset{x\to -\infty }{\lim }{e}^x \underset{x\to -\infty }{\lim }{\tan }^{-1}x \\ =e^{-\infty } {\tan }^{-1}\infty \\ =\frac{1}{e^{\infty }}\left(\frac{\mathrm{\pi }}{2}\right) \\ =\frac{1}{\infty }\left(\frac{\mathrm{\pi }}{2}\right) \\ =0\left(\frac{\mathrm{\pi }}{2}\right) \\ =0 \end{gathered}$$