Chapter 9.3, Problem 12E

To determine

Toshow: that $x^2$ grows at the same rate as the function $x^2+\sin x$ .

Explanation of Solution

Given information:

The function is $x^2+\sin x$ .

proof:if the ration of the both function at limit of $x\to \infty$ exists, then the $x^2$ grows at the same rate as the function $x^2+\sin x$ .

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{x^2+\sin x}{x^2}=\underset{x\to \infty }{\lim }1+\frac{\sin x}{x^2}\text{ [divide by }{x}^2\text{]}\\ \underset{x\to \infty }{\lim }\frac{x^2+\sin x}{x^2}=1+0\text{ [substitute]}\\ \underset{x\to \infty }{\lim }\frac{x^2+\sin x}{x^2}=1\end{array}$

Thus, $x^2$ grows at the same rate as the function $x^2+\sin x$ .