Question

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

• A. 0
• B. -1
• C. -2
• D. 1

Answer 1

Answer: (C)

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