Question

To show that g(x) grows slower than f(x), what is the result of the limit as x goes to infinity of the quotient of g of x and f of x

Answer 1

There are several assumptions that we must do in order to answer this question.

First of all, we must assume that both functions grow to infinity. In fact, if $f(x) \to l_f$ and $g(x) \to l_g$ with $l_f,l_g < \infty$, you simply have

$\displaystyle \lim_{x \to \infty} \dfrac{g(x)}{f(x)} = \dfrac{l_g}{l_f}$

Secondly, we must assume that $g(x)$ grows asymptotically slower than $f(x)$. Otherwise, you might choose

$g(x) = x,\ f(x) = x+1 \implies g(x) < f(x) \forall x$

but you would have

$\displaystyle \lim_{x \to \infty} \dfrac{g(x)}{f(x)} = 1$

If instead $g(x)$ is asymptotically slower than $f(x)$, by definition of being asymptotically slower you have

$\displaystyle \lim_{x \to \infty} \dfrac{g(x)}{f(x)} = 0$