Evaluate the following limit: _ x 0 5^x-1 4+x -2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{5^\text{x}-1}{\sqrt{4+\text{x}}-2}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{5^\text{x}-1}{\sqrt{4+\text{x}}-2}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(5^\text{x}-1\big){\big(\sqrt{4+\text{x}}+2}\big)}{\big(\sqrt{4+\text{x}}-2\big)\big(\sqrt{4+\text{x}}+2\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(5^\text{x}-1\big){\big(\sqrt{4+\text{x}}+2\big)}}{\text{x}}$
$=4\text{log}5$

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