$\lim a^x x$ when $x$ approaches $\infty$ If $|a|<1$, show that
$$\lim_{x\to \infty} a^x x =0$$
L'Hospital's Rule doesn't work, in this case. How can I analyze the given function to conclude this ?

Hint
I suggest that you consider the different cases :$a>1$,$a=1$,$a<1$. For the first two cases, the answer seems to be obvious :$\infty$. But the problem is totally different when $a<1$.
I am sure that you can take from here.
