Thread: Limit proofs

Hi, can someone please help me with this question:

Suppose f: (0, infinity) -> R is differentiable, determine whether the following are true or false and give proofs.

(i) If the limit of f(x) (with x tending to infinity) exists and is finite, then the limit of f '(x) (with x tending to infinity) is zero.

(ii) If the limit of f '(x) (with x tending to infinity) is zero, then the limit of f(x)/x (with x tending to infinity) is zero.


I'm not even sure where to start, so any help would be great
Thank you!!

Originally Posted by pseudonym

Hi, can someone please help me with this question:

Suppose f: (0, infinity) -> R is differentiable, determine whether the following are true or false and give proofs.

(i) If the limit of f(x) (with x tending to infinity) exists and is finite, then the limit of f '(x) (with x tending to infinity) is zero.

(ii) If the limit of f '(x) (with x tending to infinity) is zero, then the limit of f(x)/x (with x tending to infinity) is zero.

I'm not even sure where to start, so any help would be great

Not wanting to give the game away by just telling you the answer, for (i) think geometrically: can a function get very small (as ) and yet oscillate so rapidly that its derivative does not get small? (See attachment.)

Hint for (ii): l'Hôpital.

Originally Posted by Opalg

Not wanting to give the game away by just telling you the answer, for (i) think geometrically: can a function get very small (as ) and yet oscillate so rapidly that its derivative does not get small? (See attachment.)

Hint for (ii): l'Hôpital.

I understand the problem geometrically but the problem that I am facing is trying to construct a proof. Could someone help me on this.

Oh I have an additional problem to this question:
(iii) If the limit of f"x) (with x tending to infinity) exists and is finite, and the limit of f '(x) (with x tending to infinity) is b then b = 0.

Once again I think it is true and I can see it geometrically but having the trouble constructing the proof.

pseudonym, i've wrote something on your wall.