Is this the right way to prove basic limit identities?

Question

For example let's say I wanted to prove that:

$$\lim_{x \to \infty} \frac{1}{x} = 0$$

Normally I would use epsilon-delta for such proofs but then I start out with conditions like:

$|\frac{1}{x} - 0| < \epsilon$

$0 < |x - \infty| < \delta$

And I don't know how we're supposed to do epsilon-delta proofs for things approaching infinity, if it even makes sense to do this.

Answer 1

Infinty creates a mess of things...

What is infinity? Loosely, it is something bigger than any natural number $N$

When you have a limit $\lim_\limits{x\to\infty} f(x) = L$

You get rid of the $\delta$ in your definition and you replace it with something based on $N$

$\forall \epsilon>0\exists N>0: x>N \implies |f(x) - L| <\epsilon$

If you want to prove

$\lim_\limits{x\to a} f(x) = \infty$

You replace $\epsilon$ with $M$

$\forall M>0\exists \delta>0: |x-a|<\delta \implies |f(x)| >M$

Nothing has fundmentally changed from the definitions, it is just that we expect $\epsilon$ and $\delta$ to be small things and $M, N$ to be big things.

As far as a picture, How about this?

Answer 2

Note that in this case the definition of limit requires to find an $\bar x$ such that $$\forall x > \bar x \implies|\frac{1}{x} - 0| < \epsilon$$

Thus in this case, as for the sequences, you don't have a "small" $\delta$ but a "big" $\bar x$ for which the condition holds.