A) \[-\frac{3}{2}\]
B) \[\frac{1}{2}\]
C) \[\frac{3}{2}\]
D) None of these
Correct Answer: A
Solution :
\[f(x)={{\cot }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)=\frac{\pi }{2}-3{{\tan }^{-1}}x\] and \[g(x)={{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)=2{{\tan }^{-1}}x\] \[\therefore \] \[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)-f(a)}{g(x)-g(a)}\] \[=\underset{x\to a}{\mathop{\lim }}\,\frac{\frac{\pi }{2}-3{{\tan }^{-1}}x-\frac{\pi }{2}+3{{\tan }^{-1}}a}{2{{\tan }^{-1}}x-2{{\tan }^{-1}}a}\] \[=\frac{3}{2}\underset{x\to a}{\mathop{\lim }}\,\frac{{{\tan }^{-1}}x-{{\tan }^{-1}}a}{{{\tan }^{-1}}x-{{\tan }^{-1}}a}\] \[=-3/2\]
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