# Skipping a calculus topic (squeeze theorem)

Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.

We are on limits and L’Hôpital’s Rule, and I see this among the questions (note, all the worksheet questions are meant to be solved via L-H rule) - The instruction is

"Evaluate the following using L’Hôpital’s Rule"

$$\lim_{x\to 0}\frac{\sin x}x=$$

I recall, when subbing for a calc teacher, that this is a classic example of the use of the "squeeze theorem" aka "sandwich theorem". Once it's proven, we'd go on to different arguments of Sine, practice a bit, then move on. It's introduced prior to L-H rule.

Given the fast pace of my student self-studying and effort of remote teaching, I'm inclined to ignore this, and move on. My question is whether skipping over Squeeze Theorem is doing her a disservice, and should I (forgive the pun) squeeze it into our next session? At my HS, students have told me it feels like it's introduced, practiced for a few problems, but never seeing again.

• Can you elaborate a bit on what the actual mathematical question is? There needs to be some natural (English) language direction or question attached to that limit statement. Jul 21, 2022 at 17:02
• I think @DanielR.Collins is requesting that you clarify what question the student is being asked, that you're asking for advice on. That is, $\lim {\sin x\over x}=$ is not a question... and certainly lacks context... so could you provide that context (a little more specific than "second semester calculus", since that can mean many different things). Jul 21, 2022 at 18:03
• I know this is unrelated to the main question, but instructing students to evaluate $\lim_{x \to 0} \frac{\sin x}{x}$ using de l'Hospital's rule is (in my opinion) counterproductive. If the student already knows the derivative of $\sin x$, it'd better for them to try and recognize the definition of the derivative here, instead of blindly following a computational rule. Jul 21, 2022 at 19:06
• Yes, Michal, this is the situation I am often in. As a tutor, I need to follow what the teacher wants. While in the school, I'm careful not to offer a criticism of a teacher's methods. For outside tutoring, I'm still careful to avoid negative comments towards their teacher. Here, I'm just trying to get feedback on the one thing that feels missing, 'squeeze theorem'. Jul 21, 2022 at 19:40
• As @paulgarrett noted, I'm looking for the direction or question posed to the student. If the worksheet direction was, "evaluate each of the limits below", then that should be included in your post. As a separate issue, writing "$a =$" is a sentence fragment and a huge sore spot for some of us. Students incorrectly reading "=" as if it's a question is specifically one of the major broken understandings that some of us spend long years trying to remediate. Jul 21, 2022 at 20:10

The very limit you have chosen as an example shows the necessity of the squeeze theorem.

When deriving the fact that the derivative of sine is cosine, you first use the angle sum formula for sine, which then reduces the computation to the derivative of sine at 0. This is the limit you have chosen as an example.

So you really have no choice: to convince someone that the derivative of sine is cosine, you need the squeeze theorem. You can, of course, just skip the justification but that defeats the point of a mathematics class for me.

• I am old and dense. Are you just agreeing the teacher skipped over an important topic, or suggesting that I take the time to teach it to my student? Jul 21, 2022 at 19:42
• Depends on your goals. Will the squeeze theorem help this student score well on their exams? Probably not. Will it help them to see that mathematics is a deductive art, and that the facts they are learning have reasoning (not just authority or experimental evidence) behind them? Yes. Jul 21, 2022 at 19:45
• There is more than one road that leads to Rome. This also goes for deriving the fact that the derivative of a sine is a cosine, and not all roads involve explicit limit statements. For example, once the link between Trig functions and exponentials has been established, deriving this requires just the product and chain rules. Jul 22, 2022 at 7:22
• @TimRias generally the trig functions are "defined" geometrically in precalculus. So the pathway I suggest is pretty standard. They can be defined by power series (in which case showing derivative of sine is cosine is easy). They can be defined by differential equations in which case this fact is baked into the definition. They could be defined in terms of the complex exponential, in which case it is just the chain rule as you say. However, actually understanding complex functions is way beyond a calculus student in their 3rd week of study. Jul 22, 2022 at 9:19

The squeeze theorem merely introduces a basic framework for future analytic thinking. For example, the inequality

$$\sum_{k=1}^n\frac1k>\int_1^{n+1}\frac{1}{x}\, dx$$

can be established easily by drawing boxes that fit above the curve $$y=\frac1x$$.

Upon knowing that the integral goes to infinity like $$\ln(x)$$, we use "squeeze theorem thinking" to deduce that the harmonic series diverges: the series is squeezed between $$\ln(n+1)$$ and infinity as $$n\to\infty$$.

The point is that introducing the squeeze theorem sets a basic principle. Then with neuroplasticity, one can reuse and invoke the basic concept in future contexts, perhaps without even realizing explicitly that we are using it!

The rationale for including the squeeze theorem in Calculus I is not to narrowly give the learner a tool to evaluate certain limits per se. Applying the squeeze theorem to limits such as $$\lim_{x\to0}\frac{\sin(x)}{x}$$ is merely action that helps settle the abstract concept so that it can grow in the brain to become a schema. Analysts use the squeeze theorem subconsciously in all sorts of abstract situations, but they had to develop this capacity.

• Sorry if my question is poorly worded. I understand, and can explain the process. I am trying to determine if I need to teach it to the student whose professor is skipping it completely. Jul 21, 2022 at 17:41

Most of the other answers in this thread focus on the mathematics. This is appropriate, as this is a Q&A site for mathematics educators. However, I suspect that the question being answered ("Should I teach the squeeze theorem?") has already been addressed here many times. The distinguishing question here seems to be "Should I, as a tutor, teach a tutee the squeeze theorem when the primary instructor has omitted it?" While this question might be more appropriate for the Academia SE, it doesn't seem off-topic here.

Typically, the role of a tutor is to provide supplemental coaching to a student who is taking a course from a primary instructor. It is the job of the primary instructor to establish the curriculum, the grading schemata, the pace, etc. The job of the tutor is to review the material presented by the primary instructor in order to prepare the student for assessments.

### Are you doing harm by omitting the squeeze theorem?

No.

As a tutor, you serve your tutees by preparing them to perform well on the assessments provided by their primary instructor. When you help them with material that is likely to be on those assessments, you are doing your job. If you spend time on material that is not likely to be on those assessments, you are, perhaps, helping the student to learn more, but may be harming them in the sense that they may perform less well on assessments. Coving extra material is, likely, neutral at best.

### Is the student being harmed by the omission of the squeeze theorem?

In my opinion, yes.

As has been pointed out in the comments and in other answers, the squeeze theorem is a fundamental result in analysis (or, perhaps more foundational, is the result that $$f(x) \le g(x)$$ implies that $$\lim f(x) \le \lim g(x)$$). Skipping this result does a disservice to the mathematics, and has the potential to create a kind of "cargo cult" version of mathematics.

Thus, in my opinion, if an instructor chooses to omit the squeeze theorem, they are doing a disservice to their students. But that is on the instructor, not you.

### What should you do?

Again, this is probably a matter of opinion, but: your role is to prepare your tutees for the assessments their instructor is likely to prepare. This instructor has skipped the squeeze theorem, hence you should probably also skip it, or spend only minimal time on it.

You also need to be careful to respect the authority of the instructor. You and the instructor should appear to be a united team in front of the students. Be careful not to criticize the instructor in earshot of your tutees. Doing so only degrades the relationship between the instructor and their students, which is likely to make the overall learning environment worse.

On the other hand, it might be worth talking to the instructor. I can imagine many possible results, including:

• Students are often unreliable reporters when it comes to describing what has been taught. Perhaps the instructor will tell you that they did spend time on the squeeze theorem, and the student has simply forgotten (in which case, you really need to spend some time on it).
• Perhaps the instructor had a really good reason for omitting the squeeze theorem. I have difficulty imagining what that reason is, but you might find that they have a persuasive argument.
• Maybe the instructor just forgot about it (e.g. if they are teaching multiple sections of the course, they may have gotten confused about what was said to one group of students vs another). They may welcome the feedback, and take it as an opportunity to fill a gap.
• Or the instructor could be a real a-hole who doesn't really know what they are doing, skipped the theorem because it is hard and students often struggle with it and they want to make sure that their teaching evaluations are good. That would suck, but such is life.

Whatever the case, having a line of communication between instructor and tutor can be valuable. If it is possible for you to talk to this instructor, I would advise you to do so.

Unless you're training a superstar, and college sophomore taking calc 1 is a clue here, I would not cover material that the regular teacher has made the choice not to cover. And there are always limits on time and ability, thus choices on what to cover.

Also, can the limit be found using lhopital? Which student has probably had? Take the derivative of top and bottom. And you get 1/cos0. Or 1/1 or 1. This lesson was specifically on lhopital. This drill sheet expects practice with a specific method.