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