Functions Question 105
Question: $ \underset{x\to 1}{\mathop{\lim }},\frac{1-\sqrt{x}}{{{({{\cos }^{-1}}x)}^{2}}}= $
[AI CBSE 1990]
Options:
A) 1
B) $ \frac{1}{2} $
C) $ \frac{1}{4} $
D) Put $ {{\cos }^{-1}}x=y $ and $ x\to 1,\Rightarrow y\to 0. $ $ \underset{x\to 1}{\mathop{\lim }}\frac{1-\sqrt{x}}{{{({{\cos }^{-1}}x)}^{2}}}=\underset{y\to 0}{\mathop{\lim }}\frac{1-\sqrt{\cos y}}{y^{2}} $ Now rationalizing it, we get $ \underset{y\to 0}{\mathop{\lim }}\frac{(1-\cos y)}{y^{2}(1+\sqrt{\cos y})} $ $ =\underset{y\to 0}{\mathop{\lim }}\frac{1-\cos y}{y^{2}},.,\underset{y\to 0}{\mathop{\lim }}\frac{1}{1+\sqrt{\cos y}}=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}. $
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Answer:
Correct Answer: D
Solution:
None of these
BETA
