Q: when you say

is a non negative number and

for

**any** positive real number, what can you conclude about

, i mean, what is a(the) possible value for

? (give me an answer..)

**Definition: **Let

and let

be a function. We say that

*limit of* *as* *approaches* is

, denoted by

if and only if for every

**(epsilon)** , there is a

**(delta)** such that

whenever

.

Remark: We shall assume that

, that is,

exists.

Remark: The

is usually in terms of

, that is

is a function of

. Hence,

depends on the value of

.

So how do we use this?

Let us take the first example:

By our first method, we concluded that the

*limit of* is 3 as

goes to 1, and we had that conclusion by observing the behavior of

. The formal way to prove it is using the epsilon-delta definition.

Proof:

So, we let

be an arbitrary positive real number. We want to find a necessary

(in terms of epsilon) such that if

, then

.

Let us take a look on

.

and we want it to be less than epsilon, that is

.

So let us take

. Then whenever

, we have

. Therefore

.

Can you still follow? Just raise your questions if you have any.

And also, prove that

.