Q. 7 - Calculus

Question

Consider the limit expression

$\underset{x\to 0}{\lim }\frac{3^x-4^x}{3^x}$

Calculate the limit.

Step-by-Step Solution

Answer

The limit  of the expression is 0.

Step 1. Given information

The  given limit expression is 

$\underset{x\to 0}{\lim }\frac{3^x-4^x}{3^x}$

Step 2. Calculation

The expression can be evaluated as follows:

$$\begin{gathered} \underset{x\to 0}{\lim }\frac{3^x-4^x}{3^x} \\ =\underset{x\to 0}{\lim }\frac{3^x-{\displaystyle \frac{4^x}{3^x}}}{} \\ =\underset{x\to 0}{\lim }\left(1\right)-\underset{x\to 0}{\lim }{\left(\frac{4}{3}\right)}^x=1-{\left(\frac{4}{3}\right)}^0 \\ =1-1 \\ =0 \end{gathered}$$