3 mark for review $lim_{x\rightarrow0}\frac{2^{x}-1}{x}=$ a 0 b $ln 2$ c 1 d $\frac{1}{ln 2}$
3 mark for review $lim_{x\rightarrow0}\frac{2^{x}-1}{x}=$ a 0 b $ln 2$ c 1 d $\frac{1}{ln 2}$
Answer
Explanation:
Step1: Recall derivative definition
The derivative of a function $y = a^x$ is $y^\prime=\lim_{x\rightarrow0}\frac{a^{x + h}-a^{x}}{h}=a^{x}\ln a$. When $x = 0$, $y^\prime=\lim_{h\rightarrow0}\frac{a^{h}-1}{h}=\ln a$. Here $a = 2$.
Step2: Identify the limit
For the limit $\lim_{x\rightarrow0}\frac{2^{x}-1}{x}$, by the above - mentioned derivative formula when $a = 2$, this limit is equal to $\ln 2$.
Answer:
B. $\ln 2$