Q. 92 - Calculus

Question

Use L’Hopital’s rule to prove that every power function

with a positive power dominates the logarithmic function $g\left(x\right)=\ln \left(x\right)$

Step-by-Step Solution

Answer

The every positive function with a positive power dominates the logarithmic function.

Step1. Given information

The given function is $g\left(x\right)=\ln \left(x\right)$

Step 2. Evaluate the value

consider the values of $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$

  $$\begin{gathered} \underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to \infty }{\lim }\frac{x^2}{\ln \left(x\right)} \\ =\underset{x\to \infty }{\lim }\frac{2x}{{\displaystyle \frac{1}{x}}} \\ =\underset{x\to \infty }{\lim }2x^2 \\ =\infty \end{gathered}$$

Step 3. The solution

The every positive function with a positive power dominates the logarithmic function.