Question

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

Answer 1

Consider the problem

$\underset{x\to 0}{lim}\frac{\sqrt{1+x}-1}{x}=\underset{x\to 0}{lim}\frac{{\left(1+x\right)}^{\frac{1}{2}}-1}{x}\left(\frac{0}{0}form\right)$

Using L-Hospital rule

$\begin{array}{c}{=lim}_{x\to 0}\frac{1}{2}{\left(1+x\right)}^{-\frac{1}{2}}\\ =\frac{1}{2}\times 1\\ =\frac{1}{2}\end{array}$