Question

The value of $\underset{\alpha \to 0}{\lim }\frac{\cos e{c}^{-1}(\sec \alpha )+{\cot }^{-1}(\tan \alpha )+{\cot }^{-1}\cos \left({\sin }^{-1}\alpha \right)}{\alpha }$ is

• A. 0
• B. -1
• C. -2
• D. 1

Answer 1

Answer: (C)

$\underset{\alpha \to 0}{\lim }\frac{+{\cot }^{-1}\cos \left({\sin }^{-1}\alpha \right)}{\alpha }$
$=\underset{\alpha \to 0}{\lim }\frac{{\text{cosec}}^{-1}\left(\text{cosec}\left(\frac{\pi }{2}-\alpha \right)\right)+{\cot }^{-1}\left(\cot \left(\frac{\pi }{2}-\alpha \right)\right)+{\cot }^{-1}\cos \left[{\cos }^{-1}\sqrt{\left(1-{\alpha }^2\right)}\right]}{\alpha }$
$=\underset{\alpha \to 0}{\lim }\frac{\frac{\pi }{2}-\alpha +\frac{\pi }{2}-\alpha +{\cot }^{-1}\sqrt{1-{\alpha }^2}}{\alpha }$
$=\underset{\alpha \to 0}{\lim }\frac{-2-\frac{1}{1+1-{\alpha }^2}\left(\frac{1}{2\sqrt{1-{\alpha }^2}}(-2\alpha )\right)}{1}$
$=-2$