lim _{x rightarrow 0} frac{tan 2x - 2tan x}{( | vedclass

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(D) Let $L = \lim _{x ightarrow 0} \frac{	an 2x - 2	an x}{(1 - \cos x)(2^x - 1)}$.Using the expansion $	an x = x + \frac{x^3}{3} + O(x^5)$:$	an

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$\frac{4}{\ln 2}$

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(D) Let $L = \lim _{x ightarrow 0} \frac{	an 2x - 2	an x}{(1 - \cos x)(2^x - 1)}$.Using the expansion $	an x = x + \frac{x^3}{3} + O(x^5)$:$	an 2x = 2x + \frac{(2x)^3}{3} + O(x^5) = 2x + \frac{8x^3}{3} + O(x^5)$.$2	an x = 2(x + \frac{x^3}{3} + O(x^5)) = 2x + \frac{2x^3}{3} + O(x^5)$.Numerator: $	an 2x - 2	an x = (2x + \frac{8x^3}{3}) - (2x + \frac{2x^3}{3}) = \frac{6x^3}{3} = 2x^3$.Denominator: $(1 - \cos x)(2^x - 1) \approx (\frac{x^2}{2})(x \ln 2) = \frac{x^3 \ln 2}{2}$.$L = \lim _{x ightarrow 0} \frac{2x^3}{\frac{x^3 \ln 2}{2}} = \frac{2 	imes 2}{\ln 2} = \frac{4}{\ln 2}$.

$\lim _{x \rightarrow 0} \frac{\tan 2x - 2\tan x}{(1 - \cos x)(2^x - 1)} = $

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