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Topic: Calculating Limits using algebra.
Replies: 18   Last Post: Aug 11, 1997 2:21 PM

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Richard Sisley

Posts: 4,189
Registered: 12/6/04
Re: Calculating Limits using algebra.
Posted: Aug 1, 1997 6:59 AM
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Some in the AP calculus might find the following interesting and even
useful. I am writing material for students which includes preparation
for the AB level AP exams reflecting the new syllabus. One topic, on p.
6 of the acorn book, is limits. The book states that students should be
able to calculate limits using algebra.

In traditional calclulus courses algebra can be used to uncover limits
in zero divided by zero cases by using such techniques as factoring, or
rationalizing numerators. I was not excited about drilling students in
algebra tricks for the purpose of dealing with some zero over zero limit
problem which might appear on the AP exam, so I tried to think of some
alternative approach.

I considered the case of the quotient of two functions, both of which
are individually differentiable at a certain number, but that number is
a zero of both functions. I began with sub-cases in which that same
number was not a zero of the derivative of both the numerator and
denominator functions.

To me the idea of a tangent line as a local linear approximator is one
of the key ideas in the new AP syllabi. For example, it is at the heart
of Euler's Method and it can be used to get at the idea of an integral
as a measure of accumulated change. I thus tried to find a way to use
the local linear approximator idea to deal with the type of limit
question I described above.

I thought of tangent lines as local stand-ins for non-linear functions.
Since it is hard to transmit math symbols over the internet, I will
describe what I did in words. I hope it will be clear enough so that
others can follow with just a little writing.

Recall that the numerator function has the limit number (the number at
which the limit is sought), call it "a," as a zero. Write an expression
describing the local linear approximator of the numerator function at
the point with first coordinate "a." The denominator function also has
"a" as a zero, so write another expression describing the local liner
approximator of the denominator function at the point with first
coordinate "a." As you do this keep in mind that "a" is a zero of both
these functions, so the point-slope form is especially easy to use.

Now form a quotient of the local linear approximators. Of course this
is also a zero over zero quotient at "a." However, think about the
graph of this quotient--it is a horizontal line with a hole at the point
with first coordinate "a." Using a geometric understanding of limits
that the students are supposed to develop, the limit of the quotients of
the linear approximators at "a" is easy to "see"--graph this horizontal
line. Then look at the expression for this limit in terms of tangent
line slopes.

Some might think it would be sufficient to introduce L'Hospital's Rule
directly, even though it is no longer a stated topic in the syllabus.
But consider the above as a way to discover L'Hospital's Rule by making
use of the idea of local linear approximators and a geometric
understanding of limits.


Richard Sisley

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