Q. 21 - Calculus

Question

Calculate each of the limits in Exercises 21–48. Some of these limits are made easier by L’Hopital’s rule, and some are not 

$\underset{x\to 3}{\lim }\frac{2^x-8}{3-x}$

Step-by-Step Solution

Answer

The value of $\underset{x\to 3}{\lim }\frac{2^x-8}{3-x}$$=-8\ln 2$

Step 1. Given information

The given limit$\underset{x\to 3}{\lim }\frac{2^x-8}{3-x}$

Step 2. Using L’Hopital’s rule find the value of given limit

The limit using L’Hopitals rule is given below:

$$\begin{gathered} \underset{x\to 3}{\lim }\frac{2^x-8}{3-x}=\underset{x\to 3}{\lim }\frac{2^x\ln 2}{0-1} \\ =-\ln 2\underset{x\to 3}{\lim }{2}^x \\ =-\ln 2\times 2^3 \\ =-8\ln 2 \end{gathered}$$

Therefore,the value of the limit $-8\ln 2$