__MY ATTEMPT AT PROVING THE LIMIT__
By the definition of a limit, we know that:

Means that for every:

,theres is a

such that:

implies:

The above inequality implies that:

Also, since:

Then it follows that (by multiplying

to both sides of the g(x) inequality):

[I don't know if the follwing is neccesary for this proof, but better safe then sorry]

(We need not worry about

being negitive; since

for all

, then the denominator is always positive. And since, by the stipulations made by the problem,

for all

obviously means that

for all

; then it folllows that both the denominator and numerator are positive, therefore

for all

)

To prove that the limit approaches infinity, we must show that:

Implies that, for any

we have:

.

But we are already there! (atleast I think, as long as there are no errors) From the given limit of g(x), we were able to show that :

So letting:

and

and rememebering the stipulations made by the problem of the value of f(x), then the train of implications goes like this:

Remeber that:

So:

Which shows that:

For:

and

FINNALY, this all means that:

(granted that the stipulations on f(x) are given)