Evaluate the Limit of the Quotient of two exponential functions

Question

Say you have the following function:

$$\frac{a^x}{b^x}$$

and want to evaluate it's limit as $x$ approaches infinity. Intuition tells me that the limit will approach $0$ if $b > a$ and will approach $\infty$ if $a>b$ and plugging various equations into a graphing calculator bears this out.

However I am at a loss for how to definitively prove that this is the case. Is there a specific identity or theorem for evaluating such a limit?

Answer 1

In THIS ANSWER, I showed using only the limit definition of the exponential function and Bernoulli's Inequality that the exponential function satisfies the inequalities

$$\bbox[5px,border:2px solid #C0A000]{1+x\le e^x \le \frac {1}{1-x}} \tag 1$$

for $x<1$.

Noting that $\frac{a^x}{b^x}=e^{x\log(a/b)}$, application of $(1)$ reveals that

$$\begin{align} \bbox[5px,border:2px solid #C0A000]{1+x\log(a/b)\le \frac{a^x}{b^x}\le \frac{1}{1-x\log(a/b)} }\tag 2 \end{align}$$

for $x\log(a/b)<1$.

If $a=b$, then we see that $\lim_{x\to \infty}\frac{a^x}{b^x}=1$.

If $a>b$, then the left-hand side inequality yields $\lim_{x\to \infty}\frac{a^x}{b^x}=\infty$.

If $a<b$, then the right-hand side inequality yields $\lim_{x\to \infty}\frac{a^x}{b^x}=0$.

Answer 2

Hint:

write it as: $$ \lim_{x \to \infty}\left(\frac{a}{b} \right)^x=\lim_{x \to \infty} e^{x (\ln a-\ln b)} $$

Answer 3

Hint: what happens when $a/b<1$? What about $a/b=1$? What about.....