Question

qυєsтıση :-
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★ Evaluate -
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$\: \: \: \: \: \: \: \: \bullet\bf\: \: \: {\lim_{h \rightarrow 0} \dfrac{1}{h} \bigg{ \dfrac{1}{\sqrt{x + h}} - \dfrac{1}{\sqrt{x}} \bigg} }$
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Answer 1

$\;\;\;\bullet\;\;\;\displaystyle\lim_{\sf{h\to 0}}{\sf{\dfrac{1}{h}\bigg\{\dfrac{1}{\sqrt{x+h}}-\dfrac{1}{\sqrt{x}}\bigg\}}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\displaystyle=\lim_{\sf{h\to 0}}{\sf{\dfrac{\dfrac{1}{\sqrt{x+h}}-\dfrac{1}{\sqrt{x}}}{h}}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\displaystyle=\lim_{\sf{h\to 0}}{\sf{\dfrac{1}{h\sqrt{x+h}}}-\dfrac{1}{h\sqrt{x}}}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\displaystyle=\lim_{\sf{h\to 0}}\sf{\dfrac{\sqrt{x}}{h\sqrt{x+h\sqrt{x}}}}-\dfrac{\sqrt{x+h}}{h\sqrt{x+h}\sqrt{x}}$

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$\displaystyle=\lim_{\sf{h\to 0}}\sf{\dfrac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x(x+h)}}}×\dfrac{\sqrt{x}+\sqrt{x+h}}{\sqrt{x}+\sqrt{x+h}}$

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$\displaystyle=\lim_{\sf{h\to 0}}\sf{\dfrac{x-(x+h)}{(h\sqrt{x(x+h)})\;(\sqrt{x}+\sqrt{x+h})}}$

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$\sf{=\dfrac{-1}{2x\sqrt{x}}}$