How can i prove this interesting limit Was just playing around any thought it was interesting.
knowing now that $a>b>0$
$\lim\limits_{x\rightarrow \infty} (a^x-b^x)^{\frac{a+b}{x}} = a^{a+b}$

It can be done as follows:
$$
\lim_{x\to\infty} \left(a^x-b^x\right)^{\frac{a+b}{x}}=\lim_{x\to\infty} a^{a+b}\cdot\left(1-\left(\frac{b}{a}\right)^x\right)^{\frac{a+b}{x}}=
$$
Now we have $\lim_{x\to\infty} \left(\frac{b}{a}\right)^x=0$ and $\lim_{x\to\infty} \frac{a+b}{x}=0$ and therefore:
$$
\lim_{x\to\infty} a^{a+b}\cdot\left(1-\left(\frac{b}{a}\right)^x\right)^{\frac{a+b}{x}}=a^{a+b}\cdot\left(1-0\right)^0=a^{a+b}
$$
