1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limit of a function

  1. Sep 12, 2008 #1
    Hi
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Limit of a function
  1. Limits Of Functions (Replies: 1)

  2. Limit of function (Replies: 1)

  3. Limits of functions (Replies: 6)

  4. Limit of a function (Replies: 5)

Loading...