Q. $\lim_{x \rightarrow 0}\left\lbrack \frac{2^{x} - 1}{\sqrt{1 + x} - 1} \right\rbrack$ is equal to
A) $\log_{e}2$
B) $\log_{e}\sqrt{2}$
C) $\log_{e}4$
D) 2
Explanation
(c)
$\lim_{x \rightarrow 0}\left\lbrack \frac{2^{x} - 1}{\sqrt{1 + x} - 1} \right\rbrack = \lim_{x \rightarrow 0}\frac{2^{x}\log_{e}2}{\frac{1}{2\sqrt{1 + x}}}$ [by L’ Hospital’s rule]
$= 2\log_{e}{2 = \log_{e}4}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.