# Limits of functions at infinity

1. Oct 4, 2009

### Juggler123

Suppose that a function f:R to (0,infinity) has the property that f(x) tends 0 as x tends to infinity. Prove that 1/f(x) tends to infinity as x tends to infinity.

I don't really know where to start with this problem, I'm assuming it will involve some sort of epsilon proof but that's all I know. Please any help would be great!

2. Oct 4, 2009

Write out what it means when you say that f(x) tends to zero as x tends to infinity, and think about what that means for 1/f(x).

3. Oct 4, 2009

### Juggler123

If f(x) tends to zero as x tends to infinity it means that as x gets progressively larger f(x) gets smaller and smaller thus meaning that 1/f(x) gets larger and lerger but this a proof.

4. Oct 4, 2009

No, write out the meaning of f(x) going to zero in "epsilon" form. You have the right outline in your second post, but it isn't a proof ("hand waving" would be the description). Start with "$$\lim_{x \to \infty} f(x) = 0$$ means that for any $$\epsilon > 0$$ ..." and continue.