Question

$\lim\limits _{x \rightarrow 0}\left(1-\frac{1}{2^{x}}\right)\left(\frac{1}{\sqrt{\tan x+4}-2}\right)=$

• A. $\log 16$
• B. does not exist
• C. $3 \log 2$
• D. $6 \log 2$

Answer 1

Answer: (A)

$G . E=\lim\limits _{x \rightarrow 0}\left(\frac{2^{x}-1}{2^{x}}\right)\left(\frac{\sqrt{\tan x+4}+2}{\tan x+4-4}\right)$
$=\lim\limits _{x \rightarrow 0}\left(\frac{2^{x}-1}{x}\right) \frac{x}{\tan x}\left(\frac{\sqrt{\tan x+4}+2}{2^{x}}\right)$
$=\log 2.1 .4=4 \log 2=\log 16$