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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 21 of 72
Marks: +1, -0
lim⁡x→0(csc⁡x−cot⁡x)\lim\limits_{x\to 0}(\csc x - \cot x)
Solution:  
We have, lim⁡x→0(csc⁡x−cot⁡x)\lim\limits_{x\to 0}(\csc x - \cot x)
= lim⁡x→0[1−cos⁡xsin⁡x]\lim\limits_{x\to 0}\left[\frac{1-\cos x}{\sin x}\right] = lim⁡x→01−1+2sin⁡2(x/2)2sin⁡(x/2)cos⁡(x/2)\lim\limits_{x\to 0}\frac{1-1+2\sin^2(x/2)}{2\sin(x/2)\cos(x/2)}
= lim⁡x→0sin⁡(x/2)cos⁡(x/2)\lim\limits_{x\to 0}\frac{\sin(x/2)}{\cos(x/2)} = lim⁡x→0tan⁡(x/2)\lim\limits_{x\to 0}\tan(x/2) = 0.
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