mathop {lim }limits_{x to 0} frac{{x({2^x} - | vedclass

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(B) दिया गया सीमा: $L = \mathop {\lim }\limits_{x 	o 0} \frac{{x({2^x} - 1)}}{{1 - \cos x}}$मानक सीमा $\mathop {\lim }\limits_{x 	o 0} \frac{{{2^x}

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$\log 4$

Vedclass · Answer

(B) दिया गया सीमा: $L = \mathop {\lim }\limits_{x 	o 0} \frac{{x({2^x} - 1)}}{{1 - \cos x}}$मानक सीमा $\mathop {\lim }\limits_{x 	o 0} \frac{{{2^x} - 1}}{x} = \log 2$ और $1 - \cos x = 2\sin^2(\frac{x}{2})$ का उपयोग करते हुए:$L = \mathop {\lim }\limits_{x 	o 0} \frac{{x({2^x} - 1)}}{{2\sin^2(\frac{x}{2})}}$$L = \mathop {\lim }\limits_{x 	o 0} \left( \frac{{{2^x} - 1}}{x} ight) \cdot \left( \frac{x^2}{2\sin^2(\frac{x}{2})} ight)$$L = (\log 2) \cdot \mathop {\lim }\limits_{x 	o 0} \frac{1}{2 \left( \frac{\sin(x/2)}{x} ight)^2}$चूंकि $\mathop {\lim }\limits_{x 	o 0} \frac{\sin(x/2)}{x/2} = 1$,इसलिए $\mathop {\lim }\limits_{x 	o 0} \frac{\sin(x/2)}{x} = \frac{1}{2}$:$L = (\log 2) \cdot \frac{1}{2 \cdot (1/2)^2} = (\log 2) \cdot \frac{1}{2 \cdot (1/4)} = (\log 2) \cdot 2 = 2\log 2 = \log 4$.

$\mathop {\lim }\limits_{x \to 0} \frac{{x({2^x} - 1)}}{{1 - \cos x}} = $

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