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Limit of a function

  1. Oct 16, 2012 #1
    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..
     
  2. jcsd
  3. Oct 16, 2012 #2

    arildno

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    No.
    Because a continous 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.
     
    Last edited: Oct 16, 2012
  4. Oct 16, 2012 #3
    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..
     
  5. Oct 16, 2012 #4
    Also if i call a function to be one to one then this should hold , right?
     
  6. Oct 16, 2012 #5

    arildno

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    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)
     
  7. Oct 16, 2012 #6
    Thanks..
     
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