Experience First

In this lesson students learn how to evaluate limits from rational functions by identifying the locations of holes and vertical asymptotes. Students grapple with the idea of getting “0/0” when using direct substitution. As you are monitoring students working on question 3a,  ask them what 0/0 means. When they answer, play devil’s advocate: “I thought 0 divided by anything is zero. And isn’t something divided by 0 always undefined? And isn’t anything divided by itself equal to 1? So which one is it? Undefined? 0? 1? Something else?” This is the whole idea behind an indeterminate form--it doesn’t always have the same value in every situation.

We hope that students’ memories of our Unit 2 lessons on rational functions are activated in this lesson! Continue to challenge students to explain why the graph has a hole at x=2 and a vertical asymptote of x=-1. Ultimately, we want students to be able to say more than just “factor shows up in numerator and denominator” and “factor shows up only in the denominator”. This is of course a helpful starting point but it does not reveal the whole story.

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Formalize Later

This lesson builds on ideas from Unit 2 about rational function. While students should already know how to identify holes and vertical asymptotes by factoring the numerator and denominator, it takes time to build the conceptual understanding of the difference between holes and vertical asymptotes and how this is related to the factors of the numerator and denominator function.

When debriefing question 3b, we like to say that the value of the limit tells us what should have happened at x=2, if there was not a hole. This relates back to the idea of a limit being an intended value. It also helps explain how the “simplified form” in 5a is identical to the original function EXCEPT that it is defined at x=2, and thus we can see what the y-value should have been (aka, the y-value of the hole).

Make sure that students know that “0/0” is not the value of the limit! It is simply an indicator  that the limit is indeterminate with the current strategy of direct substitution, and another strategy will need to be found. In AP Calculus, it will become very important that students never write “=0/0” since 0/0 is not a fixed value that can be equivalent to another fixed value. Although I use correct notation when I write my work, I tend to overlook some of these mistakes when my students make them initially. Insisting on the rigor of AP Calculus when students first encounter an idea at the Precalculus level is demotivating and takes away from the purpose of the lesson.

One idea that is obvious to us teachers but less transparent to teachers is the relationship between the factored form of a rational function and the result of using direct substitution to evaluate the limit. While students know that a common factor in the numerator and denominator denotes a hole, and they learn that the indeterminate form 0/0 indicates a hole, they don’t necessarily see the relationship between the two, namely that the numerator and denominator functions have the same zero!