Evaluate the given limit: mathop {lim }limits | vedclass

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(A) At $x=0,$ the given function takes the indeterminate form $\infty - \infty$.$\mathop {\lim }\limits_{x 	o 0} (	ext{cosec}\,x - \cot x) = \mathop

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(A) At $x=0,$ the given function takes the indeterminate form $\infty - \infty$.$\mathop {\lim }\limits_{x 	o 0} (	ext{cosec}\,x - \cot x) = \mathop {\lim }\limits_{x 	o 0} \left(\frac{1}{\sin x} - \frac{\cos x}{\sin x}ight)$$= \mathop {\lim }\limits_{x 	o 0} \left(\frac{1 - \cos x}{\sin x}ight)$Using the half-angle identities $1 - \cos x = 2\sin^2(x/2)$ and $\sin x = 2\sin(x/2)\cos(x/2)$:$= \mathop {\lim }\limits_{x 	o 0} \frac{2\sin^2(x/2)}{2\sin(x/2)\cos(x/2)}$$= \mathop {\lim }\limits_{x 	o 0} 	an(x/2)$$= 	an(0) = 0$

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} (\text{cosec}\,x - \cot x)$

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