I know that in a $\alpha$-stable distribution we have:
$$ \lim_{x\rightarrow +\infty}f(x,\alpha,\beta)\sim -\alpha \gamma^\alpha \frac{\Gamma(\alpha)}{\pi}sin(\frac{\pi \alpha}{2})(1+\beta)x^{-(\alpha+1)} $$
and
$$ \lim_{x\rightarrow +\infty}P(X>x_0)\sim \gamma^\alpha \frac{\Gamma(\alpha)}{\pi}sin(\frac{\pi \alpha}{2})(1+\beta)x^{-\alpha} $$
so plotting $P/f$ we must have straight line at x>>1 such that
$$ \lim_{x\rightarrow+\infty}\frac{P(X>x_0)}{f(x,\alpha,\beta)}\sim -\frac{x}{\alpha} $$
If i perform the same calculation for a normal distribution i have:
$$ \lim_{x\rightarrow +\infty}\frac{P_N(X>x_0)}{f_N(x,\alpha,\beta)}\sim \frac{1}{2x} $$
Why there is such a big difference in the tails? namely if i can think a gauss distribution as an $\alpha$-stable distribution with $\alpha=2,\beta=0$ why they are so different?