Evaluate the following limit: _ h 0 x+h - x h ,x 0 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{h}\rightarrow0}\frac{\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}}}{\text{h}},\text{x}\ne0$

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Answer

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

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