If you take the definition of the limit:
You may also understand this by delta-epsilon notation:
where
and
To prove that
you need to use the definition:
where
so I can rewrite:
Now I can choose
. So I prooved, that for all epsilons exists such delta that
Another example: Proove that
.
Solution: Again, I have to use definition:
. Now I have to find such delta, that satisfies epsilon. It is easy:
There's another trick:
So now, if you tell me some epsilon, I am able to tell you delta which satisfies your epsilon. If you tell
my delta is