How to calculate limit as x approaches infinity of a^x/b^x?

Question

I'm trying to calcululate $\lim_{x\to\infty}\frac{a^x}{b^x}$.

I've done a few examples on Wolfram Alpha, and it seems if $a>b$ it goes to infinity and if $a<b$ it goes to $0$, but I am not sure how to prove it.

L'Hopital is normally what I would try, but it doesn't seem to work here because it doesn't really change the structure of the numerator or denominator.

Answer 1

Hint:

Let $r= \frac{a}b$.

The question is equivalent to \begin{align}\lim_{x \rightarrow \infty} r^x&=\lim_{x\rightarrow \infty}\exp(x\ln(r)) = \exp(\lim_{x\rightarrow \infty}x\ln(r))\end{align}

what happens if $r>1$?

What happens if $r<1$?

Answer 2

Hint: review properties of exponents, especially that the quotient of two numbers each raised to the the $x$ power is equal to the quotient of the two numbers itself raised to the $x$ power.