Finding $\lim_{x\rightarrow 0}\frac{10^x-2^x-5^x+1}{x\tan(x)}= $? How to calculate the following limit : $$\lim_{x\rightarrow 0}\frac{10^x-2^x-5^x+1}{x\tan(x)}$$
Thanks in advance.

The trick (to get a fast answer) is to write the numerator as a product:
$$10^x - 2^x - 5^x + 1 = (2^x-1)(5^x-1).$$
Since $\lim\limits_{x\to 0} \frac{\tan x}{x} = 1$, we get the result $\ln 2\cdot \ln 5$.
