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.
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$\begingroup$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?
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.
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