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

1. Jun 6, 2010

Moogie

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. Jun 6, 2010

Gekko

Its called the algebraic limit theorem

3. Jun 6, 2010

Hurkyl

Staff Emeritus
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.

4. Jun 6, 2010

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,
$$\lim_{h \to 0^+} 1/h = \infty$$
and
$$\lim_{h \to 0^-} 1/h = -\infty$$

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
$$\lim_{h \to 0} 1/h \text{does not exist}$$

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.