Chapter 9.3, Problem 49E

To determine

To calculate which function grows faster than ${\log }_{13}x$ as $x\to \infty$

Answer to Problem 49E

The function $x\ln x$ grows faster than ${\log }_{13}x$ .The correct answer is option E

Explanation of Solution

Given information:

The given options are,

A. $e^{-x}$

B. ${\log }_{2}x$

C. $\ln x$

D. $\log x$

E. $x\ln x$

When $\left(\frac{f\left(x\right)}{g\left(x\right)}=\infty \right)$ then it is said that $f\left(x\right)$ grows faster than $g\left(x\right)$

So, when $\left(\underset{x\to \infty }{\lim }\frac{x\ln x}{{\log }_{13}x}\right)$ is calculated its gives the answer as $\infty$

Here is the solution,

  $\begin{array}{l}\frac{x\ln x}{{\log }_{13}x}\\ =\underset{x\to \infty }{\lim }\frac{x\ln x}{{\log }_{13}x}\\ =\underset{x\to \infty }{\lim }\frac{\frac{x}{\ln x}}{\frac{x}{{\log }_{13}}}\\ =\underset{x\to \infty }{\lim }\frac{x}{0}=\infty \end{array}$

Therefore, $\left(\frac{x\ln x}{{\log }_{13}x}=\infty \right)$ so the function $x\ln x$ grows faster than ${\log }_{13}x$