The Squeeze Theorem

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The Squeeze Theorem

Based on “The Squeeze Theorem” in section 2.3 of our textbook and this discussion of the Squeeze Theorem.

The Basic Theorem

Suppose f(x) and h(x) are two functions that are defined everywhere near x = 1 (but not necessarily at x = 1 itself), and that wherever they are defined, f(x) ≥ h(x). Furthermore, suppose lim_{x → 1} f(x) = lim_{x → 1} h(x) = 17. How could you use the Squeeze Theorem to show that the average of f(x) and h(x) also approaches 17 as x approaches 1?

The key realization is that the average is in between f(x) and h(x), i.e., f(x) ≥ ( f(x) + h(x) ) / 2 ≥ h(x), and so the average is “squeezed” between f(x) and h(x). When their limits meet up, the average must too.

Does the Squeeze Theorem give you numbers or functions?

Since the Squeeze Theorem only talks about what happens at a specific point where two functions have the same limit, it gives you a single numeric value for a third function’s limit at that point. (Though I suppose there could be cases where two functions are equal over some interval, and you could use the Squeeze Theorem to show that a third function is also equal to those two over the same interval, but that would be a rare combination of circumstances.) In general, most limits are single numbers describing the behavior of a function as it heads towards a point, although there are exceptions such as limits used to find derivative functions.

Trigonometric Limits

Try to make sense of some of the limit derivations in the book. This is really an example of how you might use pauses and questions to yourself while reading mathematics.

We looked at the first derivation, namely lim_{Θ → 0} sinΘ = 0.

It starts by referring to Figure 2.29. There’s not too much to interpret in this sentence, but it is a good idea to find the figure right now and get a sense of what it shows.

Next the book says that “sinΘ is the y coordinate on the unit circle and it corresponds to the line segment shown in blue.” This is a place to stop and check that you understand what the sentence is saying, and that the claim makes sense to you. So go back to the figure and find the y coordinate this sentence is talking about and the blue line. It does make sense that the line’s length, and thus the y coordinate of its top end, is sinΘ, because the line forms one side of a right triangle, in particular the side opposite to angle Θ. Furthermore, what it means for the circle to be a “unit circle” is that its radius, which forms the hypotenuse of the triangle, is 1. Then from the definition of sine as “opposite over hypotenuse,” the length of the blue line is indeed sinΘ.

The book then asserts that “[t]he radian measure of angle θ is the length of the arc it subtends on the unit circle.” At this sentence, you might need to stop and just figure out what the words mean in the first place: “radian measure” is the number of radians in Θ; the “arc it subtends” is the part of the circle enclosed in the angle. Now try to rephrase the sentence in your own words, and check that it makes sense. Rephrasing, the sentence means that the part of the circle enclosed in angle Θ has length Θ (as long as you measure in radians). This is true because of the definition of “radian” as the size of the angle that subtends a length-1 part of a unit circle. (If you didn’t know that this was the definition of “radian,” it probably would have been hard to see why this sentence was true; this would then be a part of the reading to ask about, e.g., in an email to me, in an SI session, in a Canvas discussion, in class, etc.)

Now, “for 0 < θ < π/2, 0 < sinΘ < Θ.” This is definitely a claim that requires stopping and filling in for yourself why it’s a “therefore” from the previous ideas. Thinking about the graph of sine between 0 and π/2 is a good way to realize that the “0 < sinΘ” part of the claim is indeed true:

The “sinΘ < Θ” part is easiest to see in the figure, where sinΘ is a straight line (the blue one mentioned in the first sentence) between the x axis and a certain point, and Θ is the length of a curved line (the arc of the circle) between that same point and the axis. The straight line will always be shorter than the curved one, so indeed sinΘ < Θ.

It could also be tremendously helpful in giving a sense of what to expect next if you realize that the 3-part inequality 0 < sinΘ < Θ is starting to look like something the Squeeze Theorem could be applied to. I’m not sure how many readers would notice this connection at this point, but if you do, hold on to it — it’s much easier to read something if your main task is to see if it’s going the way you expect than it is to read it without expectations.

The next sentence does indeed use the Squeeze Theorem to conclude that lim_{Θ → 0+} sinΘ = 0. As usual, it’s a good idea to stop and see why this claim makes sense. The limits for the lower and upper bound (the limits of 0 and Θ, respectively) are from basic limit laws, and the fact that sinΘ falls between these bounds is what the preceding sentences established. So the claim makes sense.

It’s also important to read carefully enough to spot the “+” on the limits, and to pause to ask yourself why it’s there. The reason is that the argument that sinΘ is between 0 and Θ assumed that Θ is positive. At this point you might well predict that the book will go on to show that lim_{Θ → 0-} sinΘ is also 0, and indeed that is what happens.

The first part of the argument for the limit from the left is that “for -π/2 < Θ < 0, 0 < -Θ < π/2.” Thinking about why this is true reveals that the second part (“0 < -Θ < π/2”) is just what you get if you multiply all parts of the condition “-π/2 < Θ < 0” by -1. It’s also worth trying to gauge why this is something worth saying. The answer might not be fully clear yet, but at least the “-π/2 < Θ < 0” part corresponds to negative angles, which is what you predicted the book would talk about.

Apparently as a consequence (implied by “and hence”) of “0 < -Θ < π/2,” “0 < sin(-Θ) < -Θ.” Stopping to think about why this should be true, you can see that it’s just the argument made visually in Figure 2.29 applied to -Θ.

Next, the book claims that “[c]onsequently, 0 < -sinΘ < -Θ.” Asking where this comes from and why it is “consequently” from the preceding sentence, the claim is basically the previous one, “0 < sin(-Θ) < -Θ,” but with sin(-Θ) replaced by -sinΘ. To understand why the book can do this, recall the trigonometric identity sin(-Θ) = -sinΘ. Or do it visually, by looking back at Figure 2.29 and realizing that sin(-Θ) would be the blue line flipped around the x axis, i.e., the negative of sinΘ.

Thinking about “[i]t follows that 0 > sinΘ > Θ,” it’s just the result of multiplying all parts of the “0 < -sinΘ < -Θ” inequality by -1, so it makes sense.

Nearly done, the book says that “[a]n application of the squeeze theorem produces the desired limit.” Understanding this sentence requires making explicit what “the desired limit” is, namely that lim_{Θ → 0-} sinΘ = 0, and checking that you can imagine what the “application of the squeeze theorem” would be — almost exactly the first application, squeezing sinΘ between 0 and Θ, both of which go to 0 as Θ approaches 0.

Finally, the book concludes that “[s]ince lim_Θ→0+ sinΘ = 0 and lim_Θ→0- sinΘ = 0, lim_Θ→0 sinΘ = 0.” Pausing to think about this, the first thing to recognize is that this is the goal the reading has been working towards all along — so the first derivation is done! As usual, you should also ask yourself why the claim makes sense, and realize that it is an application of the idea that when both 1-sided limits of a function are equal, then their value is also the value of the 2-sided limit.

We spent 25 or 30 minutes talking through this in class, and it was a correspondingly long piece to read in these notes. While reading such a derivation for real by yourself probably won’t take as long, it will still take a long time, and you really should be taking all the pauses, and asking all the questions, illustrated above.

Next

Derivatives of trigonometric functions.

Read “Derivatives of the Sine and Cosine Functions” in section 3.5 of the textbook. Also be aware of the differentiation formulas for other trigonometric functions in the “Derivatives of Other Trigonometric Functions” subsection, although you don’t have to read that subsection in detail.

After reading, participate in this discussion of trigonometric derivatives.

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