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.