Question

Calculate the limits of$\underset{x\to \infty }{\lim }{2}^x{x}^{-100}$

Short answer

The limits of given function is$\infty$

Step-by-step solution

Step 1. Given information 

The given function is$\underset{x\to \infty }{\lim }{2}^x{x}^{-100}$

Step 2. Calculate the limits

$\underset{x\to \infty }{\lim }{2}^x{x}^{-100}=\underset{x\to \infty }{\lim }\frac{2^x}{x^{100}}$

Using l hospitals rule

$$\begin{gathered} =\underset{x\to \infty }{\lim }\frac{2^x\ln \left(2\right)}{100x^{99}} \\ =\underset{x\to \infty }{\lim }\frac{2^x{\left(\ln \left(2\right)\right)}^{100}}{100\times 99\times 98\times ····\times 1} \\ =\infty \end{gathered}$$

Therefore the limit is$\infty$

Step 3. The solution

The limits of given function is$\infty$