Question

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

Answer 1

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

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

$={lim}_{x\to 0}\frac{x}{log(1+x)(\sqrt{1+x}+1)}$

$={lim}_{x\to 0}\frac{1}{\frac{log(1+x)}{x}}\times {lim}_{x\to 0}\frac{1}{(\sqrt{1+x}+1)}$

$=1\times \frac{1}{2}=\frac{1}{2}$