A) \[\frac{1}{2\sqrt{x}}\]
B) \[\frac{1}{\sqrt{x}}\]
C) \[2\sqrt{x}\]
D) \[\sqrt{x}\]
Correct Answer: A
Solution :
\[\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{\sqrt{x+h}-\sqrt{x}}{h}=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{{{(\sqrt{x+h})}^{2}}-{{(\sqrt{x})}^{2}}}{h\,(\sqrt{x+h}+\sqrt{x})}=\frac{1}{2\sqrt{x}}\]. Aliter : Apply L-Hospital rule, \[\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{\sqrt{x+h}-\sqrt{x}}{h}=\underset{h\to 0}{\mathop{\lim }}\,\,\,\frac{1}{2\sqrt{x+h}}=\frac{1}{2\sqrt{x}}\].
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