Why does the tail of a Fréchet distribution decay as a power law?

The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0,\; x\leq 0\\e^{-x^{-\alpha}},\; x>0\end{cases}$$

shows a power law decay at the tail (survival):

$$1- \Phi_\alpha(x) = 1 -e^{-x^{-\alpha}}\sim x^{-\alpha}, \; x \to \infty$$

How can I prove this? I suppose it is done with a Taylor series (?)

By definition, $$f$$ and $$g$$ are asymptotically equal (as $$x$$ approaches infinity), denoted by
$$f(x) \sim g (x) \; \; (x\rightarrow \infty),$$ if
$$\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)} = 1.$$
In your case, we have ($$\alpha >0$$)
$$\lim_{x\rightarrow \infty} \frac{1-e^{-x^{-\alpha}}}{x^{-\alpha}} \stackrel{y=x^{-\alpha}}{=}\lim_{y\rightarrow 0} \frac{1-e^{-y}}{y} \stackrel{L'Hospital}{=} \lim_{y\rightarrow 0} \frac{e^{-y}}{1} = 1$$