Differentiation Question 26
Question: The value of $ \underset{x\to 2}{\mathop{\lim }}\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2} $ is
Options:
A) $ \frac{1}{8\sqrt{3}} $
B) $ \frac{1}{4\sqrt{3}} $
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \underset{x\to 2}{\mathop{\lim }}\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2} $
$ =\underset{x\to 2}{\mathop{\lim }}\frac{1+\sqrt{2+x}-3}{( \sqrt{1+\sqrt{2+x}}+\sqrt{3} )(x-2)} $
(Rationalizing) $ =\underset{x\to 2}{\mathop{\lim }}\frac{\sqrt{2+x}-2}{( \sqrt{1+\sqrt{2+x}}+\sqrt{3} )(x-2)} $
$ =\underset{x\to 2}{\mathop{\lim }}\frac{(x-2)}{( \sqrt{1+\sqrt{2+x}}+\sqrt{3} )( \sqrt{2+x}+2 )(x-2)} $ (Rationalizing) $ =\frac{1}{(2\sqrt{3})4}=\frac{1}{8\sqrt{3}} $
BETA
