lim _{x rightarrow 0} frac{sin ^{2}left(pi co | vedclass

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Question

(C) हमें $L = \lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$ का मूल्यांकन करना है।जैसे $x \rightarrow 0$,$\cos x \righta

Vedclass · Accepted Answer

$4 \pi^{2}$

Vedclass · Answer

(C) हमें $L = \lim _{x ightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} xight)}{x^{4}}$ का मूल्यांकन करना है।जैसे $x ightarrow 0$,$\cos x ightarrow 1$,इसलिए $\pi \cos^4 x ightarrow \pi$.माना $u = \pi \cos^4 x$. तब $\sin^2(u) = \sin^2(\pi - u) = \sin^2(\pi(1 - \cos^4 x))$.$x ightarrow 0$ के लिए,$1 - \cos^4 x = \sin^2 x (1 + \cos^2 x)$.अतः,$L = \lim _{x ightarrow 0} \frac{\sin^2(\pi \sin^2 x (1 + \cos^2 x))}{x^4}$.$\lim_{	heta ightarrow 0} \frac{\sin 	heta}{	heta} = 1$ का उपयोग करते हुए:$L = \pi^2 \cdot \lim_{x}$ ${ightarrow 0} \left( \frac{\sin x}{x} ight)^4 \cdot (1 + \cos^2 x)^2 = \pi^2 \cdot 1^4 \cdot (1 + 1)^2 = 4 \pi^2$.

$\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$ का मान ज्ञात कीजिए।

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