Question

3 mark for review $lim_{x
ightarrow0}\frac{2^{x}-1}{x}=$ a 0 b $ln 2$ c 1 d $\frac{1}{ln 2}$

Explanation:

Step1: Recall derivative definition

The derivative of a function $y = a^x$ is $y^\prime=\lim_{x
ightarrow0}\frac{a^{x + h}-a^{x}}{h}=a^{x}\ln a$. When $x = 0$, $y^\prime=\lim_{h
ightarrow0}\frac{a^{h}-1}{h}=\ln a$. Here $a = 2$.

Step2: Identify the limit

For the limit $\lim_{x
ightarrow0}\frac{2^{x}-1}{x}$, by the above - mentioned derivative formula when $a = 2$, this limit is equal to $\ln 2$.

Answer:

B. $\ln 2$