Question

6 mark for review $lim_{x
ightarrow0}\frac{5^{x}-1}{x}$ a 0 b $ln 5$ c $5ln 5$ d 1

Explanation:

Step1: Recall derivative definition

The limit $\lim_{x
ightarrow0}\frac{a^{x}-1}{x}$ is the derivative of the function $y = a^{x}$ at $x = 0$. The derivative of $y=a^{x}$ with respect to $x$ is $y'=a^{x}\ln a$.

Step2: Evaluate at $x = 0$

When $x = 0$, substituting into $y'=a^{x}\ln a$, we get $y'(0)=a^{0}\ln a=\ln a$. Here $a = 5$.

Answer:

B. $\ln 5$