To rescue the situation when the sequence (
This is like the analogous result for product, except the algebra is a touch trickier.
For the moment, assume that
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.