1. Limited time only! Sign up for a free 30min personal 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!

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

  1. Jun 6, 2010 #1

    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

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


    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


    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]
    [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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook