Chapter 8.3, Problem 51E

To determine

Which of the following functions grows with same rate as the function $\sqrt{x^8+x^4}$ as $x\to \infty$ .

Answer to Problem 51E

The correct option is (D) $x^4$ .

Explanation of Solution

Given information : The function is $\sqrt{x^8+x^4}$ and the functions to compare with this function are $x$ , $x^2$ , $x^3$ , and $x^4$ .

Consider the limit of the ratio of the functions $\sqrt{x^8+x^4}$ and $x^4$ .

  $\begin{array}{c}\underset{x\to \infty }{\lim }\frac{\sqrt{x^8+x^4}}{x^4}=\underset{x\to \infty }{\lim }\sqrt{\frac{x^8+x^4}{x^8}}\\ =\underset{x\to \infty }{\lim }\sqrt{1+\frac{1}{x^4}}\end{array}$

Find the value of the limit obtained.

  $\begin{array}{c}\underset{x\to \infty }{\lim }\sqrt{1+\frac{1}{x^4}}=\sqrt{1+\frac{1}{{\infty }^4}}\\ =\sqrt{1+0}\\ =1\end{array}$

As the limit of the ratio of the functions is constant but not zero, the function in the denominator, that is, $x^4$ grows with same rate as the function in the numerator, that is, $\sqrt{x^8+x^4}$

Consequently, the correct option is (D).