The definition of limit

In calculus, the ε-δ definition of a limit is a finest form of limit of a function. The definition states that a limit L of a function at a point α exist if no matter how α is approached, the value returned by the function will always approach L.

Formal definition of Epsilon Delta limit

Definition of limit. A function f is said to have a limit l as x→a written as
                      
given ε>0 however small,∃a positive real number δ(ε) such that

Meaning of x→a(x approaches a)

x→a means  but | x-a | is very small. so x→a means there exist a positive number δ>0, however small,such that 0<| x-a |<δ
or   x∈(a-δ,a+δ) and 
or   x∈(a-δ,a)∪(a,a+δ)
if x∈(a-δ,a) only, then we say that x tends to a from the left and we write it asx→a- or x→a-0. similarly when x∈(a,a+δ) only, then we say that x tends to a from right. We write it as x→a+ or x→a+0.

left limit: A function f is said to have a left limit l as x→a-, written as , if given ε>0, however small, there exist a positive real number δ(ε) such that
| f(x)- l |<ε for a-δ<x<a 

Note. δ(ε) means δ depends upon ε,

Right limit. A function f is said to have a right limit l as x→a+, written as , if given ε>0, however small, there exist a positive real number δ(ε) such that
| f(x)- l |<ε for a<x<a+δ.

For example 1 :- In the graph for function f(x) below if Jame and tom the value ε, then tom gives him the number δ such that for any a in the open interval (a-δ,a+δ), the value of f(x) lies in the interval (l-ε,l+ε). In this example, as Jame make ε smaller δ satisfying this property, which show that the limit exist.

Example 2 :- Now think of the opponent ε challenge as a vertical target around L and your δ response is a horizontal shooting range around a. What does it means for your δ to be successful? It means that whenever you stand in your shooting range (except for standing at the point a itself), and you shoot, you make it into ε target, as in the picture given.

            The small an ε your opponent choose the smaller the target, and the harder your job is. You may have to pick a correspondingly smaller δ. so this implies that no matter what ε is you can always find some δ response that work, you've won the game and prove that

                

Question 1:- By use of definition of limit, show that 

Solution:- Let f(x)=4x-5,   l=3

         | f(x)-l |=| (4x-5)-3 |=| 4x-8 |=4| x-2 |

                 Let ε>0, however small, be given 

              now | f(x)-l |<ε     whenever 4| x-2 |<ε

                i.e.,   =δ(say)

given ε>0 however small, we can find a positive number δ(ε) such that

              | f(x)-l |<ε   for    0<| x-2 |<δ