Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Small question concerning limits definitation

  1. Oct 9, 2011 #1
    Small question concerning limits' definition

    Hello,

    Every time I encounter the formal definition of a limit, namely :

    "For every ε>0 there is some δ>0 such that, for all x, if 0 < |x - a| < δ, then |f(x) - l| < ε"

    I always wonder why we need to write the left bound ( 0 < ) for the |x - a| and not for the |f(x) - l|. I'd tend to think that we should either put it for both or for neither (since it's an absolute value it's pretty obvious that it's going to be >0).

    - Maybe it's to specify that |x - a| must never equal to 0?
    - The δ inequality is intuitively >0 because of the ε one so there's no need to write it.

    I truly have no idea.

    Thank you :)
     
    Last edited: Oct 9, 2011
  2. jcsd
  3. Oct 9, 2011 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    In a sense it is a quibble.
    x=a is irrelevant to the question of what happens when x -> a.
     
    Last edited: Oct 9, 2011
  4. Oct 9, 2011 #3
    In fact, it's quite important. Consider the piecewise defined function

    f(x) = x if x≠0, and
    f(0) = 1.

    It should be clear that in this case

    [itex]\lim_{x\rightarrow 0}f(x) = 0[/itex].

    However, if we used the lower bound you asked about (0≤), this limit would be undefined, since for every ε≤1, the error is not <ε when x=0. It's vital to require x≠a, because the only limits that are really interesting are the ones where including a wouldn't work, because something "strange" happens right at a.
     
    Last edited by a moderator: Oct 10, 2011
  5. Oct 10, 2011 #4
    Indeed, the limit of a function at a point a is the behavior of the function around a, not exactly at a.
    If you put the equality, it means that you talk about the continuity of the function at a.
     
  6. Oct 10, 2011 #5
    Ok thank you very much!
     
  7. Oct 10, 2011 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I once (and only once) taught a course called "Calculus for Economics and Business Administration" using a text that had been chosen by the Business Administration Department. On one page they gave the "rules of limits":
    1) [itex]\lim_{x\to a} f(x)+ g(x)= \lim_{x\to a}f(x)+ \lim_{x\to a} g(x)[/itex]
    2) [itex]\lim_{x\to a} f(x)g(x)= [\lim_{x\to a}f(x)][\lim_{x\to a}g(x)][/itex]
    3) [itex]\lim_{x\to a} \frac{f(x)}{g(x)}= \frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}[/itex]
    provided that [itex]\lim_{x\to a} g(x)\ne 0[/itex].

    Then, on the very next page, they define the derivative of f(x), at x= a, as [itex]\lim_{h\to 0} \frac{f(a+ h)}{h}[/itex], completely ignoring the fact that none of those "rules of limits" apply- especially not the third one since the denominator does go to 0.

    You need a fourth rule:
    If f(x)= g(x) for all x except x= a, then [itex]\lim_{x\to a}f(x)= \lim_{x\to a}g(x)[/itex].
     
  8. Oct 10, 2011 #7
    To be fair, I don't think it's terribly outlandish to assume (especially in an informal setting) that each mathematical formula carries the implicit directive that it holds exactly where its terms is defined. If the limit of the denominator goes to zero, then the right-hand side of rule 3 is meaningless, and therefore rule 3 is meaningless.
     
  9. Oct 10, 2011 #8
    Re: Small question concerning limits' definition

    Simple answer: Consider a constant function.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Small question concerning limits definitation
  1. Small Limits Question (Replies: 5)

  2. Definition of Limit (Replies: 3)

Loading...