Q. 92 - Calculus

Question

Use L’Hopital’s rule to prove that every power function ˆ with a positive power dominates the logarithmic function $g\left(x\right)=\ln x$

Step-by-Step Solution

Answer

Every power functions $x^n$ dominate the logarithmic function $\ln x.$

Step 1. Given information

Exponential growth function and power function are $e^x$ and $x^n, n\in Z$ respectively.

Step 2. Calculation

Consider,

$\underset{x\to \infty }{\lim }\frac{x^n}{\ln \left(x\right)} (\frac{\infty }{\infty } form); n\in Z^{+}$

Applying Hopital's rule

$=\underset{x\to \infty }{\lim }\frac{n{x}^{n-1}}{{\displaystyle \frac{1}{x}}}$

Again Applying Hopital's rule,

$=\underset{x\to \infty }{\lim }n{x}^n$

since n is positive

$$\begin{gathered} =\infty \forall x\in [0,1] \\ =\underset{x\to \infty }{\lim }\frac{x^n}{\ln \left(x\right)}=\infty \end{gathered}$$

Which is possible only if $x^n$ dominates $\ln \left(x\right).$

Thus, every power functions $x^n$ dominate the logarithmic function $\ln \left(x\right).$

Hence, theorem proved.