Understanding the Limit of a Function Definition

[A Dhingra]

hello..
I have a query about the definition of limit (the ε-δone).
" Suppose f : R → R is defined on the real line and p,L ∈ R. It is said the limit of f as x approaches p is L and written

lim _x→p f(x)=L

if the following property holds:

For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f(x) − L | < ε. "

As per the meaning of definition mentioned " given an arbitrary ε > 0,we got to find a δ > 0 small enough to guarantee that f(x) is within ε-distance from L whenever x is δ-distance of a."
Can we check for existence of a limit if we twist it a bit like this " given an arbitrary δ > 0, to find an ε> 0 small enough to guarantee that x is within δ-distance from L whenever f(x) is ε -distance of a." So here finding out an ε for a δ will allow us to say that the limit exists?

Thanks for any help..

 

[arildno]

No.
Because a continuous function f(x) can perfectly well have function values arbitrarily close to f(p), even though x is a long way from p.
For example, the function f(x)=0 is such a function. there is no epsilon that can guarantee that x is close to p, even though |f(x)-f(p)| is less than epsilon.

 

[A Dhingra]

Ahhhhh...
Just trying one example,please have a look at it and tell me if i have got exactly what you have said..
Let me consider a function f(x)=sinx in the interval [0, 4pi] then at p --> pi/2, f(x) -->1
then according to my interpretation f(x)-->1 for small region around pi/2 as well as 5pi/2.
I think i got it, the very definition of a function discards this type of limit to be true, right?

Thanks a lot..

 

[A Dhingra]

Also if i call a function to be one to one then this should hold , right?

 

[arildno]

Also if i call a function to be one to one then this should hold , right?

I believe you are right.
The criterion you set up seems to pinpoint injective, continuous functions as the class satisfying it. (There MIGHT be some subtleties here I haven't discovered that would extend the class of functions satisfying your criterion as well as the continuous, injective functions)

 

[A Dhingra]

Thanks..