U0S3V01 Limits of quotients.txt

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Let's consider limits of quotients.
We take a quotient, f over g.
If the limit of g is not zero, the limit of the quotient
is computed as you would expect.
On the other hand, if the limit of g is 0,
the formula does not work.
Dividing by 0 is not allowed.
However, if both the top and bottom tend to 0,
something very interesting happens.
The numerator and denominator are
competing against each other.
This is exactly what happens when
we compute the slope of the tangent line at A.
The slope is the limit of the quotient of the rise
over the run of the line through the two points
as B approaches A. Both the rise and the run tend to 0.
But the ratio is meaningful and has an honest limit.
How does this work?
In this section, we'll develop some tools to explore
how this is possible.