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Evaluate \[ \lim_{x\to 0}\frac{x\cdot 2^x-x}{1-\cos x}. \]

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Whenever expressions involving $a^x-1$ and $1-\cos x$ appear near $x=0$, rewrite them using the standard limits \[ \frac{a^x-1}{x}\to \log a, \qquad \frac{1-\cos x}{x^2}\to \frac12. \] This usually avoids L'Hospital's Rule completely.

Updated On: Jul 29, 2026

$\log 2$
$\dfrac{1}{2}\log 2$
$2\log 2$
$\dfrac{1}{2\log 2}$

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The Correct Option is C

Solution and Explanation

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Evaluate \[ \lim_{x\to 0}\frac{x\cdot 2^x-x}{1-\cos x}. \]

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The Correct Option is C

Solution and Explanation

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