Question

Evaluate limit of the given function at x=0.

$f\left(x\right)=\frac{6^x+3^x-2^x-1}{x}$

Answer 1

$\begin{array}{c}Wehave\\ f\left(x\right)={lim}_{x\to 0}\frac{6^x+3^x-2^x-1}{x}\\ ={lim}_{x\to 0}\frac{6^x\log 6+3^x.\log 3-2^x\log 2}{1}\\ ={lim}_{x\to 0}{3}^x{2}^x\left(\log 3+\log 2\right)+3^x\log 3-2^x\log 2\\ =\log 6+\log 3-\log 2\\ =\log \frac{6\times 3}{2}\\ =2\log 3.\end{array}$