We saw in previous pages that limits work nicely with respect to the sum of two sequences and with respect to the product of two sequences. We look at the case for division here.
Suppose (_{
)
} and (_{
)
} are convergent sequences.
We look at the sequence
To rescue the situation when the sequence (_{ ) } does not converge to zero, we have the following proposition.
Proposition.
Let _{
0
}. Then there is
Proof.
Since _{
0
} there is
This means that
Theorem.
Let (_{
)
} and (_{
)
} be convergent sequences
with limits
Proof.
This is like the analogous result for product, except the algebra is a touch trickier.
Let sequences (_{
)
} and (_{
)
} and
Since _{
0
} there is
Subproof.
Assume that
Subproof.
Now let
Subproof.
For the moment, assume that
Let
Subproof.
Let
Subproof.
Assume
Then
Now we estimate
Each of these last two terms is less that
Thus
Thus
So
Subproof.
Thus
Division,
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.