Chapter 8.4, Problem 9QR

To determine

To prove: The functions $f\left(x\right)=4e^x-5$ and $g\left(x\right)=3e^x+7$ grows at same rate as $x\to \infty$ .

Explanation of Solution

Given information: The given functions are $f\left(x\right)=4e^x-5$ and $g\left(x\right)=3e^x+7$ .

Formula used: The function $f\left(x\right)$ grows at the same rate as the function $g\left(x\right)$ as $x\to \infty$ if $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=L\ne 0$ where $L$ is finite and non zero.

According to the L’Hospital’s rule if value of the limit, $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is of the form $\frac{0}{0},\frac{\infty }{\infty }$ and $\underset{x\to c}{\lim }\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}$ exists then, the value of limit $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is same as $\underset{x\to c}{\lim }\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}$ .

Proof:

Substitute the functions and find the limit, $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ .

  $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to \infty }{\lim }\frac{4e^x-5}{3e^x+7}$

Find the value of the limit at $x=\infty$ .

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{4e^x-5}{3e^x+7}=\frac{4e^{\infty }-5}{3e^{\infty }+7}\\ \Rightarrow \underset{x\to \infty }{\lim }\frac{4e^x-5}{3e^x+7}=\frac{\infty }{\infty }\end{array}$

Since the value of the limit is of the form $\frac{\infty }{\infty }$ , L’Hospital’s rule is applicable.

Differentiate the numerator and denominator of the function with respect to $x$ .

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{4e^x-5}{3e^x+7}=\underset{x\to \infty }{\lim }\frac{4e^x}{3e^x}\\ \Rightarrow \underset{x\to \infty }{\lim }\frac{4e^x-5}{3e^x+7}=\underset{x\to \infty }{\lim }\frac{4}{3}\end{array}$

Find the value of the obtained limit.

  $\underset{x\to \infty }{\lim }\frac{4}{3}=\frac{4}{3}$

Since the value of the limit, $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is finite and non zero, the function $f\left(x\right)$ grows at same rate as $g\left(x\right)$ .

Hence Proved.