Question

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

• A. log 2
• B. $\frac {1} {2 } log 2$
• C. 2 log 2
• D. 44563

Answer 1

Answer: (C)

2 log 2

Answer 2

$\lim_{x\to0} \frac{x-2^{x} -x}{1 -\cos x} = \lim _{x\to 0} \frac{x\left(2^{x} -1\right)}{1-\cos x}$ = $\lim _{x\to 0} \frac{x^{2}}{1 -\cos x} . \frac{2^{x} -1}{x}$ = $\lim _{x\to 0} \frac{x^{2} }{1-\cos x}.\lim _{x\to 0} \frac{2^{x} -1}{x}$ = $\lim _{x\to 0} 2\left(\frac{x/2}{\sin x/2}\right)^{2} . \log2 = 2 \log 2$