Question

Evaluate : $\underset{x\to 0}{lim}\frac{{10}^x-2^x-5^x+1}{x\tan x}$

Answer 1

${lim}_{x\to 0}\frac{{10}^x-2^x-5^x+1}{x\tan x}$

${lim}_{x\to 0}\frac{(2^x-1)(5^x-1)}{x\tan x}$

${lim}_{x\to 0}x\frac{(2^x-1)(5^x-1)}{x.x\tan x}$

${lim}_{x\to 0}\frac{(2^x-1)(5^x-1)}{x^2}\times \left(1\right)$

${lim}_{x\to 0}\frac{(2^x-1)}{x}.{lim}_{x\to 0}\frac{(5^x-1)}{x}$

$\ln 2\times \ln 5$

${lim}_{x\to 0}\frac{{10}^x-2^x-5^x+1}{x\tan x}=1.115$