Question

$\underset{x\to 0}{\lim }\frac{a^x+b^x-c^x-d^x}{x}$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{a^n+b^n-c^n-d^n}{x}\right] \\ \underset{x\to 0}{\lim }\left[\frac{a^n+b^n-2-c^n-d^n+2}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\left(a^n-1\right)+\left(b^n-1\right)-\left(c^n-1\right)-\left(d^n-1\right)}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\left(\frac{a^n-1}{x}\right)+\left(\frac{b^n-1}{x}\right)-\left(\frac{c^n-1}{x}\right)-\left(\frac{d^n-1}{x}\right)\right] \\ =\log a+\log b-\log c-\log d \\ =\left(\log a+\log b\right)-\left(\log c+\log d\right) \\ =\log \left(ab\right)-\log \left(cd\right) \\ =\log \left(\frac{ab}{cd}\right) \end{gathered}$$