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Rules of limits (breaking a larger limit into 2 smaller ones)

  1. Jun 6, 2010 #1
    Hi

    What rule or theorem of limits says that you can do this with a limit

    lim(h->0) (a+b)/h =
    lim(h->0) a/h + lim(h->0) b/h

    The book i am reading has just done something like that to a limit without saying why you can do that

    thanks
     
  2. jcsd
  3. Jun 6, 2010 #2
    Its called the algebraic limit theorem
     
  4. Jun 6, 2010 #3

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    This is just the limit law for +: the sum of two limits, if they exist and the sum is well-defined, is the limit of the sum.


    More generally, it's just that + is a continuous function.
     
  5. Jun 6, 2010 #4

    Mark44

    Staff: Mentor

    If the value of a + b isn't zero, the limit does not exist.

    Have you come across the concept of one-sided limits yet? For example,
    [tex]\lim_{h \to 0^+} 1/h = \infty[/tex]
    and
    [tex]\lim_{h \to 0^-} 1/h = -\infty[/tex]

    In the first limit, h approaches zero from the positive side. In the second limit, h approaches zero from the negative side.

    An important concept of limits is that in order for the two-sided limit to exist, both one-sided limits must exist and must be equal. In the example I gave, the two one-sided limits are different, so
    [tex]\lim_{h \to 0} 1/h \text{does not exist}[/tex]

    My point is that it might not help to split (a + b)/h into a/h + b/h if the limits of the terms on the right don't exist.
     
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