$\lim_{x\to0}\,(a^x+b^x-c^x)^{\frac1x}$ 
Given $a>b>c>0$, calculate$\displaystyle\,\,\lim_{x\to0}\,(a^x+b^x-c^x)^{\frac1x}\,$

I tried doing some algebraic manipulations and squeeze it, but couldn't get much further.

$$\lim_{x\to0}(a^x+b^x-c^x)^\frac{1}{x}=[1^\infty]=\exp\lim_{x\to 0}(a^x+b^x-c^x-1)\frac{1}{x}\boxed=\\(a^x+b^x-c^x-1)\frac{1}{x}=(a^x-c^x+b^x-1)\frac{1}{x}=c^x\cdot\frac{\left(\frac{a}{c}\right)^x-1}{x}+\frac{b^x-1}{x}\\ \boxed =\exp \lim_{x\to 0}\left(c^x\cdot\frac{\left(\frac{a}{c}\right)^x-1}{x}+\frac{b^x-1}{x} \right)=\exp\left(\ln\frac{a}{c}+\ln b\right)=\exp\left( \ln \frac{ab}{c} \right)=e^\frac{ab}{c}$$
