Chapter 8, Problem 6RWDT

To determine

To find: The function which grows faster as $x\to \infty$ , $\ln \left(x\right)$ or $\ln \left(\ln \left(x\right)\right)$ . And, then tell if there is one function that grows without bounds as $x\to \infty$ and that grows more slowly than every other function.

Answer to Problem 6RWDT

The function $\ln \left(x\right)$ grows faster than $\ln \left(\ln \left(x\right)\right)$ as $x\to \infty$ .

Explanation of Solution

Given information:

The functions $\ln \left(x\right)$ and $\ln \left(\ln \left(x\right)\right)$ .

Calculation:

Graph the two functions $\ln \left(x\right)$ and $\ln \left(\ln \left(x\right)\right)$ on the same coordinate plane.

&4msp; 

The graph shows that the function $\ln \left(x\right)$ grows faster than the function $\ln \left(\ln \left(x\right)\right)$ as $x\to \infty$ .

Consider the function of the $y=x^n$ . This increases without bounds as $x\to \infty$ . There can be infinite number of functions for $n\ge 0$ . There is not one function that grows without bounds as $x\to \infty$ . The function $\ln \left(\ln \left(x\right)\right)$ grows more slowly than the function $\ln \left(x\right)$ , and the function $\ln \left(\ln \left(\ln \left(x\right)\right)\right)$ grows more slowly than the function $\ln \left(\ln \left(x\right)\right)$ . Continuing this way, one can find infinite numbers of function which grows more slowly than the other function. Thus, there does not exist a function that grows more slowly than every other function without bounds as $x\to \infty$ .