Question

Evaluate:

$\underset{x\to 0}{lim}\frac{\sqrt{1+x}-1}{\log (1+x)}$

Answer 1

Now,

$\underset{x\to 0}{lim}\frac{\sqrt{1+x}-1}{log(1+x)}$

$=\underset{x\to 0}{lim}\frac{(1+x)-1}{\left\{log\right(1+x\left)\right\}(\sqrt{1+x}+1)}$

$=\underset{x\to 0}{lim}\left(\frac{1}{\frac{log(1+x)}{x}}\right)\cdot \underset{x\to 0}{lim}\frac{1}{(\sqrt{1+x}+1)}$

$=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{2}$.