Functions Question 12
Question: $ \underset{h\to 0}{\mathop{\lim }}\frac{\sqrt{x+h}-\sqrt{x}}{h}= $
[Roorkee 1983]
Options:
A) $ \frac{1}{2\sqrt{x}} $
B) $ \frac{1}{\sqrt{x}} $
C) $ 2\sqrt{x} $
D) $ \sqrt{x} $
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Answer:
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}} $ .
BETA
