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    abhijeetp5's Avatar
    abhijeetp5 Posts: 2, Reputation: 1
    New Member
     
    #1

    May 6, 2010, 09:40 PM
    Definition of Limits - Doubt
    Hi All,

    I was going through basic definition of limts wherein it is defined that

    Limit of a function f(x) is said to be L iff,

    |f(x) - L| > 0 whenever |x-a|>0
    Source:
    http://web.mit.edu/wwmath/calculus/limits/formal.html

    meaning when 'x' is not 'a' then limit of f(x) at 'a' and value of f(x) at 'x' are also not equal.

    consider the following example:
    f(x) = x/x

    this function is undefine at x=0;
    Limt f(x) = 1
    x->0
    here a = 0


    checking the above definition,
    consider x = 1 , x!= a hence |x-a| >0
    f(1) = 1 = L hence |f(x) - L|= 0 even if |x-a| >0

    This is confusing me. Please guide.Please Rectify if I went wrong somewhere.
    ebaines's Avatar
    ebaines Posts: 12,131, Reputation: 1307
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    #2

    May 7, 2010, 01:23 PM

    Take another look at the definition - there is nothing that requires |f(x)-L| > 0. What is says is that

    if for any positive number there is a corresponding positive number such that whenever . So for your example of f(x) = 1, the limit as x 1 is indeed 1, because |f(x)-L| is always zero, and hence is always less than any positive you care to name. Consider the case of : then you're looking for a such that |f(x) - 1| = |1 - 1| < 0.1. For example you could choose = 0.5, meaning 0.5 < x < 1.5, you can then see that |f(x)-L| is less than for that full range of x. Thus the limit is L = 1.
    galactus's Avatar
    galactus Posts: 2,271, Reputation: 282
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    #3

    May 8, 2010, 04:16 AM

    ebaines done a fine job explaining. But if I may suggest something. YouTube is loaded with math tutorials of all sorts.

    Go here and watch the video:
    YouTube - Epsilon Delta Limit Definition 1
    abhijeetp5's Avatar
    abhijeetp5 Posts: 2, Reputation: 1
    New Member
     
    #4

    May 10, 2010, 10:32 PM

    Thanks ebaines, I looked the concept as
    |f(x) - L|> Epsilon instead of
    |f(x) - L|< Epsilon

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