mathop {lim }limits_{theta to frac{pi }{2}} f | vedclass

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Question

(C) To evaluate the limit $\mathop {\lim }\limits_{	heta 	o \frac{\pi }{2}} \frac{\frac{\pi }{2} - 	heta}{\cot 	heta}$,we observe that as $	heta

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$1$

Vedclass · Answer

(C) To evaluate the limit $\mathop {\lim }\limits_{	heta 	o \frac{\pi }{2}} \frac{\frac{\pi }{2} - 	heta}{\cot 	heta}$,we observe that as $	heta 	o \frac{\pi }{2}$,the expression takes the indeterminate form $\frac{0}{0}$.Applying $L'	ext{Hospital's rule}$,we differentiate the numerator and the denominator with respect to $	heta$:$\frac{d}{d	heta} (\frac{\pi}{2} - 	heta) = -1$$\frac{d}{d	heta} (\cot 	heta) = -\csc^2 	heta$Thus,the limit becomes $\mathop {\lim }\limits_{	heta 	o \frac{\pi }{2}} \frac{-1}{-\csc^2 	heta} = \mathop {\lim }\limits_{	heta 	o \frac{\pi }{2}} \sin^2 	heta$.Substituting $	heta = \frac{\pi}{2}$,we get $\sin^2(\frac{\pi}{2}) = (1)^2 = 1$.

$\mathop {\lim }\limits_{\theta \to \frac{\pi }{2}} \frac{\frac{\pi }{2} - \theta}{\cot \theta} =$

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