Chapter 9.3, Problem 1QR

To determine

To evaluate the limit.

Answer to Problem 1QR

The value of $\underset{x\to \infty }{\lim }\frac{\ln x}{e^x}$ is 0.

Explanation of Solution

Given information:

The expression is $\underset{x\to \infty }{\lim }\frac{\ln x}{e^x}$

Calculation :

The limit is $\frac{\infty }{\infty }$ type, so use l Hopital’s Rule and take derivative of the numerator and of the denominator.

  $\underset{x\to \infty }{\lim }\frac{\ln x}{e^x}=\underset{x\to \infty }{\lim }\frac{\frac{1}{x}}{e^x}=\frac{0}{\infty }\text{=0 }\left[\underset{x\to \infty }{\lim }\frac{1}{x}=0,\underset{x\to \infty }{\lim }{e}^x=\infty \right]$

Therefore,

The value of $\underset{x\to \infty }{\lim }\frac{\ln x}{e^x}$ is 0.