Q. 31 - Calculus

Question

Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L'Hôpital's rule, and some are not.

$\underset{x\to \infty }{\lim }x{\left(\frac{1}{2}\right)}^x$

Step-by-Step Solution

Answer

The value of limit is zero

Step 1. Given information

Given function $\underset{x\to \infty }{\lim }x{\left(\frac{1}{2}\right)}^x$

Step 2: Apply L-Hospital rule and solve

Calculating, we get

$$\begin{gathered} \underset{x\to \infty }{\lim }x{\left(\frac{1}{2}\right)}^x=\underset{x\to \infty }{\lim }x·2^{-x} \\ \underset{x\to \infty }{\lim }x{\left(\frac{1}{2}\right)}^x=\underset{x\to \infty }{\lim }\frac{x}{2^x} \\ \underset{x\to \infty }{\lim }x{\left(\frac{1}{2}\right)}^x=\underset{x\to \infty }{\lim }\frac{1}{2^x\ln 2} \\ =\frac{1}{\ln 2}\underset{x\to \infty }{\lim }\frac{1}{2^x} \\ =\frac{1}{\ln 2}·0 \\ =0 \end{gathered}$$