Question

${\lim }_{x\to 0}\frac{x.2^x-x}{1-cosx}$ is equal to

• A. log 2
• B. $\frac{1}{2}log2$
• C. 2 log 2
• D. 1/2

Answer 1

Answer: (C)

${\lim }_{x\to 0}\frac{x-2^x-x}{1-\cos x}={\lim }_{x\to 0}\frac{x\left(2^x-1\right)}{1-\cos x}$
= ${\lim }_{x\to 0}\frac{x^2}{1-\cos x}.\frac{2^x-1}{x}$
= ${\lim }_{x\to 0}\frac{x^2}{1-\cos x}.{\lim }_{x\to 0}\frac{2^x-1}{x}$
= ${\lim }_{x\to 0}2{\left(\frac{x/2}{\sin x/2}\right)}^2.\log 2=2\log 2$