Evaluate the following limits:

Question:

Evaluate the following limits:

$\lim _{x \rightarrow a} \frac{\cos x-\cos a}{\cot x-\cot a}$

Solution:

$=\lim _{x \rightarrow a} \frac{\cos x-\cos a}{\cot x-\cot a}$

$=\lim _{x \rightarrow a} \frac{(\cos x-\cos a)}{\frac{\sin (a-x)}{\sin x \sin a}}$

$=\sin a \times \lim _{x \rightarrow a} \frac{\sin \left(\frac{x+a}{2}\right) \times \sin x}{\cos \left(\frac{x-a}{2}\right)}$

$=\sin ^{3} a$

$\therefore \lim _{x \rightarrow a} \frac{\cos x-\cos a}{\cot x-\cot a}=\sin a \times \sin a \times \sin a$

 

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