Suppose and are convergent sequences. We look at the sequence . Immediately we encounter a problem we didn't see for addition or multiplication: the sequence may not be well-defined as some of the terms may be zero.
To rescue the situation when the sequence does not converge to zero, we have the following proposition.
Let . Then there is such that .
Since there is such that . In particular .
This means that is at least defined for all , for some , and this is good enough to mean that the sequence is defined from some point onwards.
Let and be convergent sequences with limits and respectively, and suppose . Let the sequence be defined by . Then as .
This is like the analogous result for product, except the algebra is a touch trickier.
Let sequences and and be such that and . Assume that . We must prove that .
Since there is such that .
Assume that .
Now let be arbitrary.
For the moment, assume that and let such that and such that .
Let be arbitrary.
Then and so and .
Now we estimate using the triangle inequality and the fact :
Each of these last two terms is less that : for the first, since and ; and for the second, since .
In the case , the argument is easier: choose such that , so for all we have as you can check.
So holds in either case, or .
Thus , as required.
Division, , is not defined at , so it is meaningless to even talk about the division function at such places. For sequences and both converging to zero as , the sequence might not make sense, as might be zero for infinitely many , though in other cases this limit might exist.
Even if is always nonzero, the limit sometimes does exist and sometimes doesn't. For a case when it does exist, take and . For a case when it doesn't, take and .
The idea of
pushing functions through limits in expressions involving
plus, times, divides, etc., is an important one and is discussed in
more abstract terms in the next web page.
This idea is often called
Algebra Of Limits. I dislike the
name AOL a lot, and in any case this result is not algebra,
but analysis, as this idea applies to all sorts of functions, not
just algebraic ones. For some nice functions, like plus and times, this
operation of pushing though limits works and for others it doesn't.
Those functions for which it works are called continuous.
The definition and more examples are the subject of the next page.