Chapter 9.3, Problem 9E

To determine

To show that $x^2$ grows at the same rate as the given function.

Explanation of Solution

Given information:

The given function is $x^2+4x$ .

Formula used:

g grows at the same rate as f as $x\to \infty \text{ if }\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=L\ne 0,L\text{ is finite}$

Proof:

Here,

  $f\left(x\right)=x^2+4x,g\left(x\right)=x^2$

Now substitute $f\left(x\right)$ and $g\left(x\right)$

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{x^2+4x}{x^2}\\ \text{=}\underset{x\to \infty }{\lim }\frac{1+\frac{4}{x}}{1}\left[\text{divide numerator and denominator by }{x}^2\right]\\ \\ =\underset{x\to \infty }{\lim }\frac{1+\frac{4}{x}}{1}\\ =\frac{1+0}{1}\\ =1\end{array}$

Hence,

  $x^2$ Grows at the same rate as $x^2+4x$ .