Finding an Indeterminate Limit with L'Hôpital's Rule: Help and Explanation

[swampwiz]

I can't seem to figure out how to find this seemingly simple limit (that is shown numerically to go to 0)

lim _{x → ∞} ( x r^x )

for r < 1

This is an indeterminate form of ∞ * 0, but when I try to apply L'Hôpital's rule as

lim _{x → ∞} ( r^x / ( 1 / x ) )

I end up getting an expression of the form x² r^x, with further application of the rule generating higher and higher powers of x

I'm totally stuck!

 

[gb7nash]

You have the right idea. Try this instead:

lim _{x → ∞} (x/(1/r)^x)

 

[swampwiz]

You have the right idea. Try this instead:

lim _{x → ∞} (x/(1/r)^x)

But wouldn't that be ∞ / ∞ ? L'Hopital's rule only applies to 0 / 0

 

[gb7nash]

L'Hopital's rule only applies to 0 / 0

No it doesn't. It also applies to +- inf/inf