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

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(B) We need to evaluate $\mathop {\lim }\limits_{x 	o \frac{\pi }{2}} \frac{	an 2x}{x-\frac{\pi}{2}}$.At $x=\frac{\pi}{2}$,the expression takes the

Vedclass · Accepted Answer

$2$

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(B) We need to evaluate $\mathop {\lim }\limits_{x 	o \frac{\pi }{2}} \frac{	an 2x}{x-\frac{\pi}{2}}$.At $x=\frac{\pi}{2}$,the expression takes the indeterminate form $\frac{0}{0}$.Let $y = x - \frac{\pi}{2}$. As $x 	o \frac{\pi}{2}$,$y 	o 0$. Thus,$x = y + \frac{\pi}{2}$.The limit becomes $\mathop {\lim }\limits_{y 	o 0} \frac{	an 2(y + \frac{\pi}{2})}{y}$.Since $	an(2y + \pi) = 	an 2y$,the expression simplifies to $\mathop {\lim }\limits_{y 	o 0} \frac{	an 2y}{y}$.Using the standard limit $\mathop {\lim }\limits_{	heta 	o 0} \frac{	an 	heta}{	heta} = 1$,we multiply and divide by $2$:$\mathop {\lim }\limits_{y 	o 0} \frac{	an 2y}{2y} 	imes 2 = 1 	imes 2 = 2$.

Evaluate the given limit: $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{\tan 2x}{x-\frac{\pi}{2}}$

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