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Limit of a function

  1. Sep 12, 2008 #1
    What is the definition of limit of a function
  2. jcsd
  3. Sep 12, 2008 #2
    There are a couple of ways for defining the limit of a function f(x) say as x-->a, where a could be a real nr. or infinity.
    One of these ways is using [tex]\epsilon,\delta[/tex]

    Def.Let f(x) be a function defined in an open interval containing a. Then A is said to be the limit of the function f(x) as x goes to a, if for every [tex]\epsilon>0,\exists \delta(\epsilon)>0[/tex] such that

    [tex] |f(x)-A|<\epsilon, / / / / whenever / / / / 0<|x-a|<\delta[/tex]

    and we write it: [tex] \lim_{x\rightarrow a}f(x)=A[/tex]

    Another way to define it, is using heine's definition using sequences, then another way is in terms of infinitesimals.

    Note: the reason that it is required that [tex]0<|x-a|[/tex] is that it is not necessary for the function f to be defined at x=a, since when working with the limit as x-->a we are not really interested what happens exactly at x=a, but rather how the function behaves in a vicinity of a.
  4. Sep 13, 2008 #3
    If [tex]f[/tex] is a function and [tex]\epsilon[/tex] is an infinitesimal, the real part of [tex]f(x+\epsilon)[/tex] is the limit as f approaches x.

    It is useful for when a function with "holes" in them, as well as functions which jump up infinitely high when they are evaluated close to a point. For example,

    [tex]f(x) = \frac{x^3}{x}[/tex]

    is a function which is no defined at x=0. If you graph the function, it looks *exactly* the same as x^2, except that there is a "hole" at the origin. Since f(0) = 0^3 / 0 = 0/0, it is undefined.

    Taking the limit:

    [tex]\lim_{x->0} f(x)[/tex]

    allows us to ignore this illegal move, and give us a well-defined answer that is "for all practical purposes" equivalent.

    Limits crop up everywhere in calculus. The definition of a derivative, for example is:

    [tex]f'(x) = \lim_{h-> 0} \frac{f(x+h) - f(x)}{h}[/tex]

    If you were to try an evaluate [tex]\frac{f(x+h) - f(x)}{h}[/tex] with h = 0, you'd get an undefined answer. Taking the limit instead allows us to get a useful answer.
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