Evaluate the following limit: _ x 0 1+x -1 log(1+x) - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-1}{\text{log(1+x)}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-1}{\text{log(1+x)}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{1+\text{x}}-1\big)\big(\sqrt{1+\text{x}}+1\big)}{\log\text{(1+x)}{\big(\sqrt{1+\text{x}}+1\big)}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{x}}{\text{log(1+x)}{\big(\sqrt{1+\text{x}}+1\big)}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\frac{\text{log(1+x)}}{\text{x}}}\times\lim\limits_{\text{x}\rightarrow0}\frac{1}{\big(\sqrt{1+\text{x}+1}\big)}$
$=1\times\frac{1}{2}$
$=\frac{1}{2}$

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