Chapter 9.3, Problem 2QR

To determine

To evaluate the limit.

Answer to Problem 2QR

The value of $\underset{x\to \infty }{\lim }\frac{e^x}{x^3}=\infty$

Explanation of Solution

Given information:

The expression is $\underset{x\to \infty }{\lim }\frac{e^x}{x^3}$

Calculation :

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

  $\underset{x\to \infty }{\lim }\frac{e^x}{x^3}=\underset{x\to \infty }{\lim }\frac{e^x}{3x^2}=\underset{x\to \infty }{\lim }\frac{e^x}{6x}\text{=}\underset{x\to \infty }{\lim }\frac{e^x}{6}\text{=}\infty \left[\underset{x\to \infty }{\lim }{e}^x=\infty \right]$

Therefore,

The value of $\underset{x\to \infty }{\lim }\frac{e^x}{x^3}=\infty$