Solve $\lim_{x \rightarrow 0} (a^x + b^x - c^x)^\frac{1}{x}$ I need to solve the limit
$$\lim_{x \rightarrow 0} (a^x + b^x - c^x)^\frac{1}{x}$$
when $a,b,c \gt 0$.
I'm looking for ways to avoid $\frac{1}{x}$ power.

HINT:
$$
  \left( a^x + b^x - c^x \right)^\frac{1}{x} = a \left( 1 + \left(\frac{b}{a}\right)^x - \left(\frac{c}{a}\right)^x \right)^\frac{1}{x} = a \left( 1 + x \cdot \frac{\left(\frac{b}{a}\right)^x - \left(\frac{c}{a}\right)^x}{x} \right)^\frac{1}{x}
$$
Compute $\lim_{x \to 0} \frac{\left(\frac{b}{a}\right)^x - \left(\frac{c}{a}\right)^x}{x}$, then use the golden limit $\lim_{x\to 0} \left( 1 + \alpha x\right)^\frac{1}{x}$.
