lim _{x rightarrow 0} frac{x cot 4x}{sin ^2 x | vedclass

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Question

(B) We are given the limit $L = \lim _{x ightarrow 0} \frac{x \cot 4x}{\sin ^2 x \cot ^2(2x)}$.Using the identity $\cot 	heta = \frac{1}{	an 	het

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(B) We are given the limit $L = \lim _{x ightarrow 0} \frac{x \cot 4x}{\sin ^2 x \cot ^2(2x)}$.Using the identity $\cot 	heta = \frac{1}{	an 	heta}$,we rewrite the expression:$L = \lim _{x ightarrow 0} \frac{x 	an ^2(2x)}{\sin ^2 x 	an 4x}$.Now,we use the standard limits $\lim _{	heta ightarrow 0} \frac{	an 	heta}{	heta} = 1$ and $\lim _{	heta ightarrow 0} \frac{\sin 	heta}{	heta} = 1$:$L = \lim _{x ightarrow 0} \left[ \frac{x \cdot (	an 2x)^2}{(\sin x)^2 \cdot 	an 4x} ight] = \lim _{x ightarrow 0} \left[ \frac{x \cdot \frac{(	an 2x)^2}{(2x)^2} \cdot (2x)^2}{\frac{(\sin x)^2}{x^2} \cdot x^2 \cdot \frac{	an 4x}{4x} \cdot 4x} ight]$.Simplifying the terms:$L = \lim _{x ightarrow 0} \left[ \frac{x \cdot 1^2 \cdot 4x^2}{1^2 \cdot x^2 \cdot 1 \cdot 4x} ight] = \lim _{x ightarrow 0} \frac{4x^3}{4x^3} = 1$.

$\lim _{x \rightarrow 0} \frac{x \cot 4x}{\sin ^2 x \cdot \cot ^2(2x)}$ is equal to

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