lim _{x rightarrow 0} frac{8}{sin ^8 x} left{ | vedclass

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Question

(B) माना व्यंजक $L = \lim _{x \rightarrow 0} \frac{8}{\sin ^8 x} \left\{1-\cos \left(\frac{x^2}{2}\right)-\cos \left(\frac{x^2}{4}\right)+\cos \left(\

Vedclass · Accepted Answer

$\frac{1}{32}$

Vedclass · Answer

(B) माना व्यंजक $L = \lim _{x ightarrow 0} \frac{8}{\sin ^8 x} \left\{1-\cos \left(\frac{x^2}{2}ight)-\cos \left(\frac{x^2}{4}ight)+\cos \left(\frac{x^2}{2}ight) \cos \left(\frac{x^2}{4}ight)ight\}$ है।कोष्ठक के अंदर के व्यंजक का गुणनखंड करने पर:$1-\cos \left(\frac{x^2}{2}ight)-\cos \left(\frac{x^2}{4}ight)+\cos \left(\frac{x^2}{2}ight) \cos \left(\frac{x^2}{4}ight) = \left(1-\cos \left(\frac{x^2}{2}ight)ight) \left(1-\cos \left(\frac{x^2}{4}ight)ight)$.सर्वसमिका $1-\cos 	heta = 2 \sin ^2 \left(\frac{	heta}{2}ight)$ का उपयोग करने पर:$L = \lim _{x ightarrow 0} \frac{32 \sin ^2 \left(\frac{x^2}{4}ight) \sin ^2 \left(\frac{x^2}{8}ight)}{\sin ^8 x}$.सीमा $\lim _{x ightarrow 0} \frac{\sin x}{x} = 1$ का उपयोग करने पर:$L = 32 \cdot \frac{1}{16 \cdot 64} = \frac{32}{1024} = \frac{1}{32}$.

$\lim _{x \rightarrow 0} \frac{8}{\sin ^8 x} \left\{1-\cos \left(\frac{x^2}{2}\right)-\cos \left(\frac{x^2}{4}\right)+\cos \left(\frac{x^2}{2}\right) \cos \left(\frac{x^2}{4}\right)\right\} =$

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