$\lim a^x x$ 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 ?

Note that $\ln a ^x x = x\ln a +\ln x$. It follows that
$$
\frac{\ln a^xx}{x}\rightarrow \ln a
$$
as $x\rightarrow \infty$. Now consider what happens to $\ln a^xx$ and apply the exponential function.
