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) (g(x))/(f(x)) = (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) (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) (g(x))/(f(x)) = 0`