Can the trigonometric limit be evaluated by "induction"?

Question

Can this limit: $\displaystyle \lim_{x \to 0} \dfrac{\sin x}{x}$ be evaluated by 'induction', as in here?

My motivation for asking is this lecture note from MIT's Calculus Open Course Ware, where the professor 'proves' this limit for his students by doing this, and it left me flabbergasted as I had to learn it in a not hard, but complicated way.

As a soft follow up question, If the way used to "prove" the limit isn't correct, why the professor has chosen this way? As an HS student, I think that the class would be fully able to understand a complete proof. (Comparing the area.)

The lecture here

Disclaimer: I sure do not want to question the way that the professor teaches, I just want to understand the motivation behind it. — Deltab, Feb 09 '18 at 21:19
Induction... on what? And that "proof" you mention in the lecture notes seems to be begin as the usual, trigonometric proof...but then it derails strongly into an "intuitive" argument I don't like at all and, for me, that's not a proof at all. — DonAntonio, Feb 09 '18 at 21:19
The “proof” essentially states that the limit is $1$ because it is evidently so. — egreg, Feb 09 '18 at 22:07
@DonAntonio that is why I used induction between quotes, for I don't have a better word for this "result justification". Now would someone explain to me why he did this? — Deltab, Feb 10 '18 at 00:09
The professor gave a rather intuitive 'proof' when there are actual, understandable proofs of this limit. — Deltab, Feb 10 '18 at 00:17
Probably worth noting that this is "induction" in the epistemological sense, which has absolutely nothing to do with mathematical induction. — Micah, Feb 10 '18 at 01:22
@Micah Yeah, that was what I was trying to convey, but what could I expect by using such a word on a math forum? — Deltab, Feb 10 '18 at 04:40

score 2 · Accepted Answer · answered Feb 10 '18 at 01:14

We see from Figure 2 that as $\theta$ shrinks, the length $\sin(\theta)$ of the segment gets closer and closer to the length $\theta$ of the curved arc. We conclude that as $\theta\to0$, $\dfrac{\sin\theta}{\theta}\to 1$.

This proves nothing at all: saying that the cathetus gets closer and closer to the arc is exactly the same as saying the sought limit is $1$. This fallacy is commonly known as begging the question.

Yes, it might be “intuitively clear”, but one of the aims of mathematics is avoiding intuition, that can lead to mistakes. It's not difficult to find functions that, tested on a single precision calculator, seem to have a certain limit, but actually have another: we don't really know what happens when the angle “shrinks so much that we're not even able to make a drawing”, which is the same issue as with the kind of limit described above.

There are different levels of rigor, of course. Also the proof you advocate, with areas, suffers from similar problems: the very concept of measure of an angle is far from rigorous and so is the concept of area. But, at beginner's level, it may be used for conveying ideas how limits can be computed. With a simple comparison of areas, we deduce that, for $\theta>0$, $$ \cos\theta\le\frac{\sin\theta}{\theta}\le\frac{1}{\cos\theta} \tag{*} $$ and also that $\sin\theta\le\theta$. This latter inequality immediately provides continuity of the sine and hence of the cosine (by a translation), so (*) gives our limit by squeezing. Later on, everything can be made fully rigorous by defining the circular functions by their Taylor series.

If one wants to appeal to intuition, the “getting closer and closer” can be better derived from the isochronism of a pendulum (for small amplitudes), with physical experiments. Definitely not, in my opinion, just from looking at a picture.

+1 for the use of the phrase "begging the question". This appears to be a popular form of hand waving. — Paramanand Singh, Feb 10 '18 at 03:12
Great answer, side note: Considering that math supports itself on axioms that are preestablished before the reasoning process we can infer that math is actually circular, i.e. you always have to establish something as true. Circular reasoning isn't that bad, It only means that either conclusion is true if the other one is also true. And your axiomatic system is entitled to you, of course. — Deltab, Feb 10 '18 at 03:51

score 0 · Answer 2 · answered Feb 10 '18 at 00:43

Can this limit: $\displaystyle \lim_{x \to 0} \dfrac{\sin x}{x}$ be evaluated by 'induction', as in here?

Certainly not. Tables of values can only give us a guess, possibly wrong. For example, from the first table below "we have" $\displaystyle\lim_{x\to 0}\sin(\tfrac{1}{x})=0$ and, from the second, $\displaystyle\lim_{x\to 0}\sin(\tfrac{1}{x})=\tfrac{\pi}{4}$.

Why the professor has chosen this way?

Because it works. If a reasoning with pictures works, we call it a "geometric proof" (example). If it doesn't work, we call it a "paradox" (example).

score 0 · Answer 3 · answered Feb 10 '18 at 02:49

The OP should look at Is the proof of $\lim_{\theta\to 0} \frac{\sin \theta}{\theta}=1$ in some high school textbooks circular?

But not I just want to work on a 'warm-up' exercise.

Consider a point $(x, y)$ on the unit circle $x^2 + y^2 = 1$ with both $x$ and $y$ greater than $0$, so that we have a point in the first quadrant. Two line segments are of interest:

The line segment length $S$ between $(x, 0)$ and $(x, y)$. This is the sine of some angle, but that is not the focus here.

The line segment length $A$ between $(1, 0)$ and $(x, y)$. This is an approximation of arc length on the circle as $x$ goes to $1$.

Show that $\frac{S}{A}$ approaches $1$ as $x$ approaches $1$ from the left.

Hint: This is really elementary and all that is required is understanding some facts about continuous functions.

While working on this problem draw some pictures and think about taking the next step, looking at $\frac{sin(\theta)}{\theta}$ as $\theta$ goes to $0$.

Can the trigonometric limit be evaluated by "induction"?

3 Answers3