Q. 91 - Calculus

Question

Use L’Hopital’s rule to prove that every exponential growth function dominates the power function $g\left(x\right)=x^2$

Step-by-Step Solution

Answer

Every exponential growth function dominates the power function

Step 1. Given information

The given function is $g\left(x\right)=x^2$

Step 2. Evaluate the values

Evaluate $f\left(x\right) and g\left(x\right)$

$$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }A{e}^{kx} \\ =A{e}^{k\left(\infty \right)} \\ =\infty \\ \underset{x\to \infty }{\lim }g\left(x\right)=\underset{x\to \infty }{\lim }{x}^2 \\ ={\infty }^2 \\ =\infty \end{gathered}$$

$$\begin{gathered} \underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to \infty }{\lim }\frac{A{e}^{kx}}{x^2} \\ =\underset{x\to \infty }{\lim }\frac{A{k}^2{e}^{kx}}{2} \\ =\infty \end{gathered}$$

Step 3. The solution

Every exponential growth function dominates the power function