1
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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 $r<1$?

$\endgroup$
1
  • $\begingroup$ Can this be done without logarithms? Perhaps by the squeeze theorem? $\endgroup$ Mar 8 '21 at 4:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.