Q. 80 - Calculus

Question

Calculate the limits $\underset{x\to \infty }{\lim }{e}^{-x}\ln \left(x\right)$

Step-by-Step Solution

Answer

The limits of given function $\underset{x\to \infty }{\lim }{e}^{-x}\ln \left(x\right)$ is 0

Step 1. Given information

The given information is $\underset{x\to \infty }{\lim }{e}^{-x}\ln \left(x\right)$

Step 2. Calculate the limits

$\underset{x\to \infty }{\lim }{e}^{-x}\ln \left(x\right)=\underset{x\to \infty }{\lim }\frac{\ln \left(x\right)}{e^x}$

Using the L hospitals rule 

$$\begin{gathered} =\underset{x\to \infty }{\lim }\frac{{\displaystyle \frac{1}{x}}}{e^x} \\ =\underset{x\to \infty }{\lim }\frac{1}{x{e}^x} \\ =0 \end{gathered}$$

The limit is 0

Step 3. The solution

The limit of given function is 0