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

[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

 

[Gekko]

Its called the algebraic limit theorem

 

[Hurkyl]

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.

 

[Mark44]

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.