Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow 0}(\operatorname{cosec} x-\cot x)$

 

Solution:

$=\lim _{x \rightarrow 0}(\csc x-\cot x)$

$=\lim _{x \rightarrow 0}\left(\frac{1}{\sin x}-\frac{\cos x}{\sin x}\right)$

$=\lim _{x \rightarrow 0}\left(\frac{1-\cos x}{\sin x}\right)$

$=\lim _{x \rightarrow 0}\left(\frac{2 \times \sin \frac{x}{2} \times \sin \frac{x}{2}}{2 \times \sin \frac{x}{2} \times \cos \frac{x}{2}}\right)[\because 1-\cos \theta=2 \sin \theta \times \sin \theta]$

$=\lim _{x \rightarrow 0}\left(\tan \frac{x}{2}\right)$

$=0$

$\therefore \lim _{x \rightarrow 0}(\csc x-\cot x)=0$

 

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