Section 6.2 Limits of the form zero over zero: Please explain more clearly.

[markbarsamian]

Section 6.2 Limits of the form zero over zero: Please explain more clearly.

I my calculus courses, students are forever confused about the issue of when one can and cannot cancel factors in an expression like f(x) = (x-1)(x-2)/(x-2)

Near the top of the page, in the first example, you say note that if we assume x is not equal to 2, then we can cancel terms. I think this needs much more and much clearer explanation. Why can we "assume" that x is not equal to 2?

Here's how I do it:

In my courses, I stress to the students that the most important concept of the first month of the course is the idea that when computing f(2), one cannot cancel factors and so f(2) DNE, while when computing the limit, as x approaches 2, of f(x), one does cancel.

I compute f(1), f(2), f(3), and show how numbers in the numerator do cancel when computing f(1) and f(3). But I point out that f(2) leads to 0/0, which is undefined. One cannot cancel 0/0.

Then I discuss that when when the limit, as x approaches 2, of f(x), one is supposed to consider the values of f(x) when x is close to 2 BUT NOT equal to 2. Since x is not equal to 2, we know that x-2 is not equal to 0, so we can cancel (x-2)/(x-2). We are definitely NOT cancelling 0/0 here. We are cancelling terms whose value we don't know, except that we do know the terms are NOT zero. I tell them that I want them to be able to explain that cancellation step that clearly, because it is the most important concept of the first month of the class.

Follow the link below to a set of lecture notes for one of my class meetings. Pages 9 - 17 discuss an example similar to the one in your book. (Of course you'll write whatever you want--it's your book--but I do think that a book's discussion of this concept should be as thorough as my lecture notes.)

https://people.ohio.edu/barsamia/2019-20.1.1350/lecture.notes/Day.02.pdf