Chapter 9.3, Problem 50E

To determine

To calculate which function grows at the same rate as $e^x$ as $x\to \infty$

Answer to Problem 50E

The function $e^{x+2}$ grows at the same rate as $e^x$ .The correct answer is option C

Explanation of Solution

Given information:

The given options are,

A. $e^{2x}$

B. $e^{3x}$

C. $e^{x+2}$

D. $e^{-x}$

E. $e^{-x+1}$

When $\left(\frac{f\left(x\right)}{g\left(x\right)}\ne 0\right)$ then it is said that $f\left(x\right)$ grows at same rate as $g\left(x\right)$

So, when $\left(\underset{x\to \infty }{\lim }\frac{e^x}{e^{x+2}}\right)$ is calculated its gives the answer as $e^{-2}$

Here is the solution,

  $\begin{array}{l}\frac{e^x}{e^{x+2}}\\ =\underset{x\to \infty }{\lim }\frac{e^x}{e^{x+2}}\\ =\underset{x\to \infty }{\lim }{e}^x{e}^{-x-2}\\ =\underset{x\to \infty }{\lim }{e}^{-2}\left[\text{the limit of the constant is equal to constant}\right]\end{array}$

Therefore, $\left(\underset{x\to \infty }{\lim }\frac{e^x}{e^{x+2}}=e^{-2}\right)$ which is not equal to zero so the function $e^{x+2}$ grows at the same rate as $e^x$