526
$\begingroup$

How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.

This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated.

$\endgroup$
16
  • 74
    $\begingroup$ l'Hôpital's rule is easiest: $\lim\limits_{x\to0}\sin x = 0$ and $\lim\limits_{x\to0}x = 0$, so $\lim\limits_{x\to 0}\frac{\sin x}x = \lim\limits_{x\to 0}\frac{\cos x}1 = 1 $ $\endgroup$
    – Joren
    Oct 23, 2011 at 20:41
  • 232
    $\begingroup$ @Joren: I'm extremely curious how will you prove then that $\sin ' x = \cos x$ $\endgroup$
    – Ilya
    Oct 24, 2011 at 9:10
  • 4
    $\begingroup$ @FUZx44xl: sure, but to be fare you first prove that $\sin x\sim x$ with $x\to 0$. Geometrically $\endgroup$
    – Ilya
    Oct 24, 2011 at 15:42
  • 12
    $\begingroup$ Recent changes in what? The definition of $\lim$, of $\sin$ or of $0$? $\endgroup$
    – Asaf Karagila
    Dec 18, 2013 at 20:03
  • 3
    $\begingroup$ @FUZxxl:Exactly what was your definition by "geometrical means"? $\endgroup$
    – Platonix
    Dec 22, 2013 at 18:45

27 Answers 27

611
$\begingroup$

sinc and tanc at 0

The area of $\triangle ABC$ is $\frac{1}{2}\sin(x)$. The area of the colored wedge is $\frac{1}{2}x$, and the area of $\triangle ABD$ is $\frac{1}{2}\tan(x)$. By inclusion, we get $$ \frac{1}{2}\tan(x)\ge\frac{1}{2}x\ge\frac{1}{2}\sin(x)\tag{1} $$ Dividing $(1)$ by $\frac{1}{2}\sin(x)$ and taking reciprocals, we get $$ \cos(x)\le\frac{\sin(x)}{x}\le1\tag{2} $$ Since $\frac{\sin(x)}{x}$ and $\cos(x)$ are even functions, $(2)$ is valid for any non-zero $x$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. Furthermore, since $\cos(x)$ is continuous near $0$ and $\cos(0) = 1$, we get that $$ \lim_{x\to0}\frac{\sin(x)}{x}=1\tag{3} $$ Also, dividing $(2)$ by $\cos(x)$, we get that $$ 1\le\frac{\tan(x)}{x}\le\sec(x)\tag{4} $$ Since $\sec(x)$ is continuous near $0$ and $\sec(0) = 1$, we get that $$ \lim_{x\to0}\frac{\tan(x)}{x}=1\tag{5} $$

$\endgroup$
34
  • 87
    $\begingroup$ Do my homework! Note that $(1)$ says that for $0\le x\le\frac{\pi}{2}$ we have $0\le\sin(x)\le x$; therefore, $\lim\limits_{x\to0^+}\sin(x)=0$. Then $\cos(x)=1-2\sin^2(x/2)$ should finish the job. $\endgroup$
    – robjohn
    Oct 23, 2011 at 18:37
  • 60
    $\begingroup$ From your comment, I wasn't expecting that you could find it on your own, but "Read the question!" seemed a bit rough around the edges. $\endgroup$
    – robjohn
    Oct 23, 2011 at 19:04
  • 36
    $\begingroup$ Sorry for that. $\endgroup$
    – FUZxxl
    Oct 23, 2011 at 19:35
  • 9
    $\begingroup$ I think that your justification is slightly circular :). In an introductory calculus course, $(\sin x,\cos x)$ is probably defined to be the point we reach after traveling $x$ units counterclockwise along the unit circle from $(1,0)$. $\endgroup$
    – Mike F
    Oct 23, 2011 at 20:54
  • 8
    $\begingroup$ @robjohn: OK fair enough, I wasn't really trying to define sine and cosine so much as I was trying to point out that radians are usually introduced via the arclength of a wedge, not the area of a wedge (divided by 2). Also, I didn't have a problem with using that arclength and area are proportional to the angle. Rather I think the gap is where we use that the ratio (area of wedge)/(arc length of wedge) is exactly 1/2. $\endgroup$
    – Mike F
    Oct 23, 2011 at 22:43
142
$\begingroup$

You should first prove that for $x > 0$ small that $\sin x < x < \tan x$. Then, dividing by $x$ you get $$ { \sin x \over x} < 1 $$ and rearranging $1 < {\tan x \over x} = {\sin x \over x \cos x }$ $$ \cos x < {\sin x \over x}. $$ Taking $x \rightarrow 0^+$ you apply the squeeze theorem. For $x < 0$ and small use that $\sin(-x) = -\sin x$ so that $${\sin(-x) \over -x} = {\sin x \over x}.$$ As far as why the first inequality I said is true, you can do this completely from triangles but I don't know how to draw the pictures here.

diagram

$\endgroup$
9
  • 5
    $\begingroup$ But how to prove that $\sin x<x<\tan x$? $\endgroup$
    – FUZxxl
    Oct 23, 2011 at 16:31
  • 2
    $\begingroup$ It is in the picture. The definition of radians makes the picture above true. Maybe that is worth mentioning: this limit explicitly depends on "$x$" being measured in radians. $\endgroup$
    – tkr
    Oct 23, 2011 at 16:33
  • 3
    $\begingroup$ This is a strange picture! Normally you want the $tan(\theta)$ side to be parallel to the $sin(\theta)$ side. $\endgroup$
    – user641
    Oct 23, 2011 at 17:36
  • 17
    $\begingroup$ If you make $\tan(\theta)$ parallel then you need to make the points $S$ and $Q$ the same. For whatever reason, this is the proof I like the most because it relates the tangent line at the point on the circle to the value we call "tangent". To each his own... $\endgroup$
    – tkr
    Oct 23, 2011 at 18:22
  • 3
    $\begingroup$ Even with this picture, you could use the area inclusion principle and argue that area of triangle OPQ < Area of sector OPQ < Area of triangle OPS. This translates into $\sin \theta < \theta < \tan \theta$ which is the first line of this answer and hence, nothing strange about it. $\endgroup$ Sep 2, 2015 at 21:05
109
$\begingroup$

Usually calculus textbooks do this using geometric arguments followed by squeezing.

Here's an Euler-esque way of looking at it---not a "proof" as that term is usually understood today, but still worth knowing about.

Let $\theta$ be the length of an arc along the circle of unit radius centered at $(0,0)$, from the point $(1,0)$ in a counterclockwise direction to some point $(\cos\theta,\sin\theta)$ on the circle. Then of course $\sin\theta$ is the height of the latter point above the $x$-axis. Now imagine what happens if $\theta$ is an infinitely small positive number. Then the arc is just an infinitely short vertical line, and the height of the endpoint above the $x$-axis is just the length of the arc. I.e. when $\theta$ is an infinitely small number, then $\sin\theta$ is the same as $\theta$. It follows that when $\theta$ is an infinitely small nonzero number, then $\dfrac{\sin\theta}{\theta}=1$.

That is how Euler viewed the matter. See his book on differential calculus.

$\endgroup$
6
  • 3
    $\begingroup$ I disagree with the recent edit to my answer and I have reverted to the previous version. "Infinitesmal" means "infinitely small". $\qquad$ $\endgroup$ Jan 21, 2016 at 19:12
  • 1
    $\begingroup$ You read Euler's book ? Was it very difficult to read because of the notation and language of the time ? $\endgroup$
    – Saikat
    Feb 27, 2016 at 4:03
  • 2
    $\begingroup$ @user230452 : Just some parts of it. I wouldn't say the differences in language and notation were the challenging part. $\qquad$ $\endgroup$ Feb 27, 2016 at 18:18
  • 2
    $\begingroup$ Wouldn't one rather say that $\frac{sin \theta}{\theta}$ is infinitely close to 1 if $\theta$ is an infinitely small nonzero number? $\endgroup$
    – Sven
    Apr 14, 2017 at 13:05
  • 5
    $\begingroup$ @Sven : That is indeed how it's done in Robinson's "nonstandard analysis". There is another approach to rigorous infinitesimals in which it would be done the way Euler did it, saying that if $\theta$ is infinitely small then $\dfrac{\sin\theta}\theta = 1.$ It's called "smooth infinitesimal analysis". In that approach, the square of an infinitesimal is $0$, so we have $$ \frac{\sin\theta} \theta = \frac{\theta - \dfrac{\theta^3} 6 + \dfrac{\theta^5}{120} - \cdots \cdots} \theta = 1. $$ en.wikipedia.org/wiki/Smooth_infinitesimal_analysis $\qquad$ $\endgroup$ Apr 14, 2017 at 18:48
90
$\begingroup$

Look at this link:

http://fatosmatematicos.blogspot.com/2010/08/provas-sem-palavras-parte-20.html

Here is the picture I copied from that blog:

Copy of the picture from the Fatos Matematicos blog

$\endgroup$
3
  • 1
    $\begingroup$ +1, nice site - What is the length BC? $\endgroup$
    – NoChance
    Oct 24, 2011 at 5:34
  • 24
    $\begingroup$ What is the argument for showing that $\theta \cos \theta\le \sin \theta$? The picture isn't immediately convincing. $\endgroup$
    – Mark Viola
    Apr 15, 2015 at 5:31
  • 2
    $\begingroup$ @Dr. MV: if you rescale the circle in $\cos \theta$ units, then the original $\sin \theta$ is a scaled $\tan \theta$ which is always longer than its arc. $\endgroup$
    – Mattsteel
    May 30, 2016 at 20:38
57
$\begingroup$

I am not sure if it counts as proof, but I have seen this done by a High Schooler. enter image description here

In the given picture above, $\displaystyle 2n \text{ EJ} = 2nR \sin\left( \frac{\pi}{n } \right ) = \text{ perimeter of polygon }$.

$\displaystyle \lim_{n\to \infty }2nR \sin\left( \frac{\pi}{n } \right ) = \lim_{n\to \infty } (\text{ perimeter of polygon }) = 2 \pi R \implies \lim_{n\to \infty}\frac{\sin\left( \frac{\pi}{n } \right )}{\left( \frac{\pi}{n } \right )} = 1$ and let $\frac{\pi}{n} = x$.

$\endgroup$
3
  • 9
    $\begingroup$ This method is usually used to prove that the perimeter of a circle is $2\pi R$ using the fact $\lim\limits_{x\to 0}\frac{\sin x}{x}=1$ $\endgroup$
    – user5402
    Jul 20, 2013 at 19:15
  • 4
    $\begingroup$ Santosh, how does one prove that the perimeter of the polygon converges to the $ 2\pi$ ? $\endgroup$
    – Imago
    Feb 5, 2016 at 19:45
  • $\begingroup$ @Imago math.stackexchange.com/questions/720935/… $\endgroup$
    – user301988
    Jun 6, 2016 at 3:32
56
$\begingroup$

I claim that for $0<x<\pi/2$ that the following holds $$\sin x \lt x \lt \tan x$$ Figure 1
In the diagram, we let $OC=OA=1$. In other words, $Arc\:CA=x$ is an arc of a unit circle. The shortest distance from point $C$ to line $AO$ is line $CE=\sin x$ (because $CE\perp OA$). Another path from point $C$ to line $OA$ is arc $CA$ (which is longer than CE because it is not the shortest path). So we have at the very least $$\sin x \lt x$$ Now we need to show that line $BA=\tan x \gt x$.
Lines $AD$ and $CD$ are both tangent to arc $CA$. $CD+DA$ is longer than arc $CA$ because the set of points bound by sector $OCA$ is a subset of the set of points bound by quadrilateral $OCDA$, both of which are convex sets. This means that the perimeter of quadrilateral $OCDA$ must be longer than the perimeter of the sector $OCA$ (as per Archimedes, On the Sphere and Cylinder Book I). But both the sector and the quadrilateral both have sides $OC$ and $OA$, so we have $$CA=x<DC+DA$$ But $BD>CD$ because it is the hypotenuse in $\triangle BCD$ we have $$\tan x = BA = BD+DA\gt CD+DA \gt CA=x \gt \sin x$$

So we have $$\sin x \lt x \lt \tan x$$ $$\frac{\sin x}{x} \lt 1 \lt \frac{\tan x}{x}=\frac{\sin x}{x}\cdot\sec x$$ From this we can extract $$\frac{\sin x}{x} \lt 1$$ and $$1 \lt \frac{\sin x}{x}\cdot\sec x$$ $$\cos x \lt \frac{\sin x}{x}$$ Putting these inequalities back together we have $$\cos x \lt \frac{\sin x}{x} \lt 1$$

Because $\displaystyle\lim_{x\to 0}\cos x = 1$, by the squeeze theorem we have $$\lim_{x\to 0}\frac{\sin x}{x}=1$$

$\endgroup$
6
  • 3
    $\begingroup$ The reason that I chose an arc length proof is because most derivations that I've seen of the area of a circle (see mathopenref.com/circleareaderive.html) assume that $\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1$. Using an area based proof, to me, seems like putting the cart before the horse. $\endgroup$
    – John Joy
    Nov 9, 2014 at 15:14
  • 2
    $\begingroup$ Actually, what Archimedes assumed was that if 2 curves (both concave in the same direction) have the same endpoints and one of the curves is strictly between a line segment with the same end points and the other curve, then the length of the enclosed curve is shorter than the length of enclosing curve. See axiom #2 archive.org/stream/worksofarchimede00arch#page/4/mode/2up $\endgroup$
    – John Joy
    Feb 5, 2015 at 14:17
  • 2
    $\begingroup$ OK, so he essentially axiomatizes the idea that among all curves connecting endpoints $A$ and $B$ satisfying some kind of regularity condition - concavity in the same direction, we have this inequality based on enclosure. This makes $arc(CA)<CD+DA$ in your figure merely equivalent to the assumption itself, but it makes sense and is very nicely put by Archimedes! Thank you for pointing me to that! $\endgroup$
    – String
    Feb 5, 2015 at 14:52
  • 2
    $\begingroup$ If you don't have a copy of Archimedes's book, you can approximate sector OCA with a polygon inside OCDA and use cut-the-knot.org/m/Geometry/PerimetersOfTwoConvexPolygons.shtml $\endgroup$ Jun 6, 2017 at 16:06
  • 2
    $\begingroup$ I love this proof! $\endgroup$
    – explogx
    Dec 4, 2019 at 0:43
53
$\begingroup$

Here you may see an elementary approach that starts from a very interesting result, see this problem. All you need is a bit of imagination. When we take $\lim_{n\rightarrow\infty} \frac{n\sin(\frac{\pi}{n})}{1+\sin(\frac{\pi}{n})}$ we may notice that we have infinitely many circles surrounding the unit circle with infinitely small diameters that lastly perfectly approximate the length of the unit circle when having it there infinity times . Therefore when multiplying n by the radius under the limit to infinity we get π. Let's denote $\frac{\pi}{n}$ by x.

$$\lim_{x\rightarrow0}\frac{\pi\sin(x)}{x(1+\sin(x))}={\pi}\Rightarrow\lim_{x\rightarrow0}\frac{\sin(x)}{x(1+\sin(x))}=1\Rightarrow\lim_{x\rightarrow0}\frac{\sin(x)}{x}=1$$

The proof is complete.

$\endgroup$
1
  • 2
    $\begingroup$ this assumes the a priori knowledge of the existence of the limit $\lim_{x\to 0}\frac{\sin(x)}{x}$, how would you go about proving it before hand? (+1) by the way for the original and alternative proof $\endgroup$
    – user153330
    Feb 24, 2016 at 14:07
47
$\begingroup$

Don't you feel strange about why most of the proofs are done with a figure? I've had this problem in the beginning, and realized after that this is due to the definition we use for the function $\sin x$. Because the usual definition of $\sin x$ we all study first in high schools depends on “classical geometry” and usually with a figure, you should depict out the figure and to make it clear.

However, if you use other definitions of $\sin x$ that are equivalent to the former, you'll find it more simple. For example,

$$\sin x = \frac{x^1}{1!} - \frac{x^3}{3!}+ \frac{x^5}{5!} - \cdots + \cdots - \cdots$$

and hence

$$\frac{\sin x}x = \frac{x^0}{1!} - \frac{x^2}{3!}+ \frac{x^4}{5!} - \cdots$$

which obviously tends to $1$ as $x$ approaches 0.

$\endgroup$
4
  • 9
    $\begingroup$ Indeed, it's easy to see that this holds if one uses a series, but this question starts on the prerequisite that one does not use a series. $\endgroup$
    – FUZxxl
    Mar 13, 2015 at 17:54
  • 4
    $\begingroup$ And how do you know that the derivative of an infinte series is equal to the sum of the derivatives of each term? $\endgroup$ Apr 16, 2016 at 4:32
  • 11
    $\begingroup$ @StevenGregory: Where does he use that? He only divides by $x$. What he does use is that $\frac{1}{x}\lim_{n\to\infty} \sum_{k=0}^n a_k = \lim_{n\to\infty}\sum_{k=0}^n \frac{a_k}{x}$ $\endgroup$
    – celtschk
    Sep 24, 2016 at 9:36
  • 4
    $\begingroup$ @celschk starting with the infinite series as a 'definition' causes more problems than it solves. Now we have to prove that this sine behaves like the sine we learned in high school. This proof is compelling but it is not really a proof. $\endgroup$ Sep 24, 2016 at 14:44
34
$\begingroup$

Here's one more: $$ \lim_{x \to 0} \frac{\sin x}{x}=\lim_{x \to 0} \lim_{v \to 0}\frac{\sin (x+v)-\sin v}{x}\\ =\lim_{v \to 0} \lim_{x \to 0}\frac{\sin (x+v)-\sin v}{x}=\lim_{v \to 0}\sin'v=\lim_{v\ \to 0} \cos v=1 $$

$\endgroup$
2
  • 47
    $\begingroup$ Usually, this limit is used to compute the derivative of $\sin(x)$. $\endgroup$
    – robjohn
    Jul 20, 2013 at 18:43
  • 3
    $\begingroup$ This is an interesting alternative, but it does raise the question of why the limiting operations can be swapped. $\endgroup$
    – Allawonder
    Jun 24, 2019 at 3:27
29
$\begingroup$

It depends on your definition of the sine function. I would suggest checking out the geometric proof in ProofWiki.

$\endgroup$
0
26
$\begingroup$

The strategy is to find $\frac{d\arcsin y}{dy}$ first. This can easily be done using the picture below.

arcsin as area

From the above picture, $\arcsin y$ is twice the area of the orange bit. The area of the red bit is ${1 \over 2}y\sqrt{1-y^2}$. The area of the red bit plus the orange bit is $\int_{0}^y \sqrt{1-Y^2} dY$. So $$\arcsin y = 2\int_{0}^y \sqrt{1-Y^2} dY - y\sqrt{1-y^2}.$$ Differentiating with respect to $y$ gives $\frac{d\arcsin y}{dy} = \frac{1}{\sqrt{1-y^2}}$. Using the theorem for the derivative of inverse functions yields $\sin' \theta = \sqrt{1 - \sin^2 \theta} = \cos \theta$.

(A similar thing can be done with the arc length definition of $\arcsin$.)

$\endgroup$
1
  • 2
    $\begingroup$ While this is indeed an interesting approach, integrals haven't been taught at the point where this limit is proved. Thank you for your answer though. $\endgroup$
    – FUZxxl
    Apr 11, 2015 at 17:38
25
$\begingroup$

Here is a different approach that uses the integral definition of the arcsine function. We will deduce the limit of interest without appeal to geometry or differential calculus.

Instead, we only rely on elementary analysis of continuous functions and their inverses along with simple properties of the Riemann integral. To that end, we now proceed.


We define the sine function, $\sin(x)$, as the inverse function of the function $f(x)$ given by

$$\bbox[5px,border:2px solid #C0A000]{f(x)=\int_0^x \frac{1}{\sqrt{1-t^2}}\,dt }\tag 1$$

for $|x|< 1$.

NOTE: It can be shown that the sine function defined as the inverse of $f(x)$ given in $(1)$ has all of the familiar properties that characterize the circular function $\sin(x)$.

It is straightforward to show that since $\frac{1}{\sqrt{1-t^2}}$ is positive and continuous for $t\in (-1,1)$, $f(x)$ is continuous and strictly increasing for $x\in (-1,1)$ with $\displaystyle\lim_{x\to 0}f(x)=f(0)=0$.

Therefore, since $f$ is continuous and strictly increasing, its inverse function, $\sin(x)$, exists and is also continuous and strictly increasing with $\displaystyle \lim_{x\to 0}\sin(x)=\sin(0)=0$.


From $(1)$, we have the bounds (SEE HERE)

$$\bbox[5px,border:2px solid #C0A000]{1 \le \frac{f(x)}x\le \frac{1}{\sqrt{1-x^2}}} \tag 2$$

for $x\in (-1,1)$, whence applying the squeeze theorem to $(2)$ yields

$$\lim_{x\to 0}\frac{f(x)}{x}=1 \tag 3$$


Finally, let $y=f(x)$ so that $x=\sin(y)$. As $x\to 0$, $y\to 0$ and we can write $(3)$ as

$$\lim_{y\to 0}\frac{y}{\sin(y)}=1$$

from which we have

$$\bbox[5px,border:2px solid #C0A000]{\lim_{y\to 0}\frac{\sin(y)}{y}=1}$$

as was to be shown!


NOTE:

We can deduce the following set of useful inequalities from $(2)$. We let $x=\sin(\theta)$ and restrict $x$ so that $x\in [0,1)$. In addition, we define new functions, $\cos(\theta)=\sqrt{1-\sin^2(\theta)}$ and $\tan(\theta)=\sin(\theta)/\cos(\theta)$.

Then, we have from $(2)$

$$\bbox[5px,border:2px solid #C0A000]{y\cos(y)\le \sin(y)\le y\le \tan(y)} $$

which are the familiar inequalities often introduced in an introductory geometry or trigonometry course.

$\endgroup$
7
  • 2
    $\begingroup$ This is similar to my answer. You're formalizing the arc-length definition of $\sin$ in the following three steps. 1) Defining $\arcsin$ geometrically 2) Expressing that informal definition using integration 3) Defining $\sin$ as the inverse of $\arcsin$. Also, where you used the sqeeuze theorem, you could equally have used the Fundamental Theorem of Calculus and the Inverse Function Theorem. Then you'd get my answer $\endgroup$
    – wlad
    Feb 5, 2018 at 10:16
  • 1
    $\begingroup$ Another difference is you used the arc-length definition and I used the area definition $\endgroup$
    – wlad
    Feb 5, 2018 at 10:19
  • 2
    $\begingroup$ @ogogmad I don't use any geometrical argument. A function $f$ is defined without reference to the arc length interpretation. Both $f$ and its inverse has properties. It can be shown that the inverse function is the well-known sine function. And while one can apply the FOC instead of the squeeze theorem, there is no advantage in doing so. So, why on earth are you leaving these comments? $\endgroup$
    – Mark Viola
    Feb 5, 2018 at 14:50
  • 1
    $\begingroup$ Because I'm pointing out that your abstract definitions make geometric sense, whether or not your realize it. I'm also pointing out that your squeeze argument is just a special case of FOC $\endgroup$
    – wlad
    Feb 5, 2018 at 16:43
  • 1
    $\begingroup$ In fact, all of the arguments above are morally the same, AFAICT. The squeeze arguments are just unpacking the proof of FOC on a special case. The FOC itself is proved using a very similar squeeze argument to the one people are using. The areas of the sectors are simply the integrals that are being differentiated. In tkr's argument, he's instead differentiating an arclength integral, which gives the same result anyway. There's the remaining feature that our two answers uses $\arcsin$ while everyone else's uses $\sin$; they may be implicitly unpacking the proof of the inverse function theorem. $\endgroup$
    – wlad
    Feb 5, 2018 at 17:01
23
$\begingroup$

Let $f:\{y\in\mathbb{R}:y\neq 0\}\to\mathbb{R}$ be a function defined by $f(x):=\dfrac{\sin x}{x}$ for all $x\in \{y\in\mathbb{R}:y\neq 0\}$.

We have $\displaystyle\lim_{x \to 0}\dfrac{\sin x}{x}=1$ if and only if for every $\varepsilon>0$, there exists a $\delta>0$ such that $|f(x)-1|<\varepsilon$ whenever $0<|x-0|<\delta$.

Let $\varepsilon>0$ be an arbitrary real number.

Note that $\sin x=\displaystyle \sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n+1}}{(2n+1)!}$.

If $x \neq 0$, we have $\dfrac{\sin x}{x}=$$\displaystyle \sum_{n=0}^{\infty}(-1)^n\dfrac{x^{2n}}{(2n+1)!}=1+$$\displaystyle \sum_{n=1}^{\infty}(-1)^n\dfrac{x^{2n}}{(2n+1)!}$.

We thus have

$|f(x)-1|=\left|\dfrac{\sin x}{x}-1\right|=\left|\displaystyle \sum_{n=1}^{\infty}(-1)^n\dfrac{x^{2n}}{(2n+1)!}\right|\leq \left|\displaystyle\sum_{n=1}^{\infty} \dfrac{x^{2n}}{(2n+1)!}\right|\leq \displaystyle\sum_{n=1}^{\infty} \left|\dfrac{x^{2n}}{(2n+1)!}\right|$

Therefore we have

$|f(x)-1|\leq \displaystyle\sum_{n=1}^{\infty} \left|\dfrac{x^{2n}}{(2n+1)!}\right|\leq \displaystyle \sum_{n=1}^{\infty} |x^{2n}|=\sum_{n=1}^{\infty}|x^2|^n$

If $0<|x|<1$, then $0<|x^2|<1$, and the infinite series $\displaystyle\sum_{n=1}^{\infty}|x^2|^n$ converges to $\dfrac{x^2}{1-x^2}$.

Choose $\delta:=\sqrt{\dfrac{\varepsilon}{1+\varepsilon}}$. Then $0<|x-0|<\delta$ implies that $0<|x|<$$\sqrt{\dfrac{\varepsilon}{1+\varepsilon}}<1$, and hence $x^2<\varepsilon-\varepsilon x^2$. But $x^2<\varepsilon-\varepsilon x^2$ implies that $\dfrac{x^2}{1-x^2}<\varepsilon$.

We therefore have $\sum_{n=1}^{\infty}|x^2|^n<\varepsilon$ whenever $0<|x-0|<\delta$. But since $|f(x)-1|\leq\displaystyle\sum_{n=1}^{\infty}|x^2|^n$, we have $|f(x)-1|<\varepsilon$ whenever $0<|x-0|<\delta$.

Since $\varepsilon$ was arbitrary, we have $\displaystyle\lim_{x \to 0}\dfrac{\sin x}{x}=1$.

$\endgroup$
1
  • 2
    $\begingroup$ The premise of the question was not to use power series. $\endgroup$
    – FUZxxl
    Dec 4, 2018 at 12:35
21
$\begingroup$

Usual proofs can be circular, but there is a simple way for proving such inequality.

Let $\theta$ be an acute angle and let $O,A,B,C,D,C'$ as in the following diagram: enter image description here

We may show that:

$$ CD \stackrel{(1)}{ \geq }\;\stackrel{\large\frown}{CB}\; \stackrel{(2)}{\geq } CB\,\stackrel{(3)}{\geq} AB $$

$(1)$: The quadrilateral $OCDC'$ and the circle sector delimited by $O,C,C'$ are two convex sets. Since the circle sector is a subset of the quadrilateral, the perimeter of the circle sector is less than the perimeter of the quadrilateral.

$(2)$: the $CB$ segment is the shortest path between $B$ and $C$.

$(3)$ $CAB$ is a right triangle, hence $CB\geq AB$ by the Pythagorean theorem.

In terms of $\theta$ we get: $$ \tan\theta \geq \theta \geq 2\sin\frac{\theta}{2} \geq \sin\theta $$ for any $\theta\in\left[0,\frac{\pi}{2}\right)$. Since the involved functions are odd functions the reverse inequality holds over $\left(-\frac{\pi}{2},0\right]$, and $\lim_{\theta\to 0}\frac{\sin\theta}{\theta}=1$ follows by squeezing.


A slightly different approach might be the following one: let us assume $\theta\in\left(0,\frac{\pi}{2}\right)$.
By $(2)$ and $(3)$ we have $$ \theta \geq 2\sin\frac{\theta}{2}\geq \sin\theta $$ hence the sequence $\{a_n\}_{n\geq 0}$ defined by $a_n = 2^n \sin\frac{\theta}{2^n}$ is increasing and bounded by $\theta$. Any increasing and bounded sequence is convergent, and we actually have $\lim_{n\to +\infty}a_n=\theta$ since $\stackrel{\large\frown}{BC}$ is a rectifiable curve and for every $n\geq 1$ the $a_n$ term is the length of a polygonal approximation of $\stackrel{\large\frown}{BC}$ through $2^{n-1}$ equal segments. In particular

$$ \forall \theta\in\left(0,\frac{\pi}{2}\right), \qquad \lim_{n\to +\infty}\frac{\sin\left(\frac{\theta}{2^n}\right)}{\frac{\theta}{2^n}} = 1 $$ and this grants that if the limit $\lim_{x\to 0}\frac{\sin x}{x}$ exists, it is $1$. By $\sin x\leq x$ we get $\limsup_{x\to 0}\frac{\sin x}{x}\leq 1$, hence it is enough to show that $\liminf_{x\to 0}\frac{\sin x}{x}\geq 1$. We already know that for any $x$ close enough to the origin the sequence $\frac{\sin x}{x},\frac{\sin(x/2)}{x/2},\frac{\sin(x/4)}{x/4},\ldots$ is convergent to $1$, hence we are done.

Long story short: $\lim_{x\to 0}\frac{\sin x}{x}=1$ follows from the fact that a circle is a rectifiable curve, and a circle is a rectifiable curve because it is the boundary of a convex, bounded subset of $\mathbb{R}^2$. The convexity of the disk follows from the triangle inequality: a disk is a closed ball for the euclidean distance.


$(1)$ relies on this powerful Lemma:

Lemma. If $A,B$ are convex bounded sets in $\mathbb{R}^2$ and $A\subsetneq B$, the perimeter of $A$ is less than the perimeter of $B$.

Proof: by boundedness and convexity, $\partial A$ and $\partial B$ are rectifiable, with lengths $L(A)=\mu(\partial A),\,L(B)=\mu(\partial B)$. Always by convexity, there is some chord in $B$ that does not meet the interior of $A$ (a tangent to $\partial A$ at a smooth point does the job, for instance). Assume that such chord has endpoints $B_1, B_2 \in \partial B$ and perform a cut along $B_1 B_2$: both the area and the perimeter of $B$ decrease, but $B$ remains a bounded convex set enclosing $A$. Since $A$ can be approximated through a sequence of consecutive cuts, $L(A)<L(B)$ follows.

$\endgroup$
3
  • 2
    $\begingroup$ Late to the party, but I think this is the non-circular (no puns intended) way of thinking that has been missing so far. Nice job! $\endgroup$
    – String
    Jun 13, 2017 at 9:05
  • 1
    $\begingroup$ Up voting! I think Archimedes used a weaker form of your Lemma to get the circumference of a circle. $\endgroup$ Jun 21, 2017 at 23:49
  • $\begingroup$ Great answer (+1) $\endgroup$
    – MathLover
    Jan 25, 2018 at 23:07
17
$\begingroup$

Let $\sin(x)$ is defined as solution of $\frac{d^2}{dx^2}\textrm{f}(x)=-\textrm{f}(x)$ with $\mathrm f(0)=0,\,\frac{d}{dx}\mathrm f(0)=C$ initial conditions, so exact solution is $\mathrm f(x)=C\cdot\sin(x)$. Define second derivative as $$ \begin{align*} \frac{d^2}{dx^2}\textrm{f}(x)=\lim_{\Delta x\to 0}{\frac{\frac{\mathrm f(x)-\mathrm f(x-\Delta x)}{\Delta x}-\frac{\mathrm f(x-\Delta x)-\mathrm f(x-2\cdot\Delta x)}{\Delta x}}{\Delta x}}&=\\=\lim_{\Delta x\to 0}{\frac{\mathrm f(x)-2\cdot \mathrm f(x-\Delta x)+\mathrm f(x-2\cdot\Delta x)}{\Delta x^2}} \end{align*} $$ we can easy check this limit for any (?) functions. Similarly, we can define the first derivative for right, middle and left points: $$ \frac{d}{dx}\textrm{f}(x)\left\{ \begin{aligned} &=\lim_{\Delta x\to 0}{\frac{\mathrm f(x)-\mathrm f(x-\Delta x)}{\Delta x}} \\ &=\lim_{\Delta x\to 0}{\frac{\mathrm f(x-\Delta x)-\mathrm f(x-2\cdot\Delta x)}{\Delta x}}\\ &=\lim_{\Delta x\to 0}{\frac{\mathrm f(x)-\mathrm f(x-2\cdot\Delta x)}{2\cdot\Delta x}} \end{aligned} \right. $$ Let's use the finite elements method assuming $Td=\Delta x,\,y_n=\mathrm f(x),\,y_{n-1}=\mathrm f(x-\Delta x),\,y_{n-2}=\mathrm f(x-2\cdot \Delta x)$ Override differential equation as $$ \frac{y_n-2\cdot y_{n-1}+y_{n-2}}{Td^2}=-y_n $$ Now solve this implicit equation for $y_n$ to obtain explicit recurrence relation: $$ y_n = \frac{2\cdot y_{n-1}-y_{n-2}}{1+Td^2} $$ Using arbitrarily small but non-zero quantity Td we can plot exponentially decaying sampled sine function (because the poles are inside the unit circle of the transfer function corresponding to the given recurrence relation). Similarly we write three systems for the initial conditions:

$$ \left\{ \begin{aligned} &y_n = \frac{2\cdot y_{n-1}-y_{n-2}}{1+Td^2} \\ &C=\frac{y_n-y_{n-1}}{Td} \end{aligned}\right. $$ $$ \left\{ \begin{aligned} &y_n = \frac{2\cdot y_{n-1}-y_{n-2}}{1+Td^2} \\ &C=\frac{y_{n-1}-y_{n-2}}{Td} \end{aligned}\right. $$ $$ \left\{ \begin{aligned} &y_n = \frac{2\cdot y_{n-1}-y_{n-2}}{1+Td^2} \\ &C=\frac{y_n-y_{n-2}}{2\cdot Td} \end{aligned}\right. $$ Solve this sequence of equations for $y_{n-1}$ and $y_{n-2}$: $$ \left\{ \begin{aligned} &y_{n-1} = -C\cdot Td + y_{n}\\ &y_{n-2}=-2\cdot C\cdot Td + y_{n}\cdot\left(1-Td^2\right)\ \end{aligned}\right. $$ $$ \left\{ \begin{aligned} &y_{n-1} = -C\cdot Td + y_{n}\cdot\left(1+Td^2\right)\\ &y_{n-2}=-2\cdot C\cdot Td + y_{n}\cdot\left(1+Td^2\right)\ \end{aligned}\right. $$ $$ \left\{ \begin{aligned} &y_{n-1} = -C\cdot Td + y_{n}\cdot\left(1+\frac{Td^2}{2}\right)\\ &y_{n-2}=-2\cdot C\cdot Td + y_{n}\ \end{aligned}\right. $$ At zero point $y_n=\mathrm f(0)=0$ and we can see linear dependence: $$ \begin{aligned} &y_{n-1} = -C\cdot Td\\ &y_{n-2}=-2\cdot C\cdot Td \end{aligned} $$ for all three solutions. Replace back: $$ \begin{array}{l} \mathrm f(0)&=0\\ \mathrm f(0-\Delta x) &= -C\cdot \Delta x\\ \mathrm f(0-2\cdot \Delta x) &= -2\cdot C\cdot \Delta x \end{array} $$ So all three $\frac{d}{dx}\mathrm f(0)$ limits is equal to $C$ at $x=0$ and in accordance with $\mathrm f(x)=C\cdot\sin(x)$ by definition we can write $$ \lim_{\Delta x\to 0}{\frac{\mathrm f(0)-\mathrm f(0-\Delta x)}{\Delta x}}=\lim_{\Delta x\to 0}{\frac{0-(-C \cdot \Delta x)}{\Delta x}}=C $$ Thus $$ \lim_{\Delta x\to 0}{\frac{\sin(0)-C\cdot\sin(0-\Delta x)}{\Delta x}}=\lim_{\Delta x\to 0}{\frac{C\cdot\sin(\Delta x)}{\Delta x}}=C\cdot\lim_{\Delta x\to 0}{\frac{\sin(\Delta x)}{\Delta x}}=C $$ and $\lim_{\Delta x\to 0}{\frac{\sin(\Delta x)}{\Delta x}}=1$

$\endgroup$
2
  • 1
    $\begingroup$ While this is an exceptionally neat answer, it is not really appropriate for the level (high school) on which I asked the question, given that computing this limit is an introductory exercise into differentials. Plus we defined $\sin$ geometrically (i.e. though the relationship between angles and sides in a right triangle), so your prerequisites must be established first. $\endgroup$
    – FUZxxl
    Aug 19, 2016 at 20:40
  • 3
    $\begingroup$ @FUZxxl do you realize that users with different levels and mathematical background may check this question? The answer given above might be helpful for them. $\endgroup$
    – Xam
    Sep 23, 2017 at 3:38
11
$\begingroup$

Simple one is using sandwich theorem Which demonstrated earlier.In this method you have to show that $\frac{\sin x}{x} $ lies between other two functions. As $x \longrightarrow 0$ both of them will tends to ONE.

Then as in the case of sandwich (if both the bread part go to one stomach the middle portion will also go to the same stomach) $\frac{\sin x}{x}$ will go to ONE.

You can use geogebra to see the visualization of this phenomena using geogebra.First you input $\sin x$ and $x$ and observe that near to $0$ values of $\sin x$ and $x$ are same.

Secondly input $\frac{\sin x}{x}$ then observe function is approaching to $1$ as $x$ tends to $0$

$\endgroup$
1
  • 8
    $\begingroup$ This describes the Sandwich Theorem, but does not answer the question. At best, this should be a comment to the question, $\endgroup$
    – robjohn
    Dec 13, 2014 at 13:31
10
$\begingroup$

Originally posted on the proofs without words post, here is a simple image that explains the derivative of $\sin(x)$, which as we all know, is directly related to the limit at hand.

enter image description here

If one is not so convinced, take a look at the above picture and notice that if $u\pm h$ is in the first quadrant, then

$$\frac{\sin(x+h)-\sin(x)}h<\cos(x)<\frac{\sin(x-h)-\sin(x)}{-h}$$


Notice that

$$ \begin{align}\frac{d}{dx}\sin(x)&=\lim_{h\to0}\frac{\sin(x+h)-\sin(x)}h\\\text{picture above}&=\lim_{h\to0}\frac{\sin(x)\cos(h)+\cos(x)\sin(h)-\sin(x)}h\\\cos(x)&=\lim_{h\to0}\sin(x)\frac{\cos(h)-1}h+\cos(x)\frac{\sin(h)}h\\\cos(0)&=\lim_{h\to0}\frac{\sin(h)}h\end{align} $$

$\endgroup$
4
  • 2
    $\begingroup$ Note that your reason is circular: In order to prove the derivative of $\sin$, we need to know the limit of $\sin h\over h.$ Attempting to prove the limit the other way round is counterproductive. $\endgroup$
    – FUZxxl
    Nov 16, 2016 at 0:47
  • 2
    $\begingroup$ @FUZxxl no actually, the whole point of this was a geometric proof of the fact. $\endgroup$ Nov 16, 2016 at 17:23
  • $\begingroup$ @FUZxxl If you are still unsatisfied about it, there is a whole post on the topic. $\endgroup$ Nov 17, 2016 at 0:44
  • $\begingroup$ @FUZxxl Since you still seem unhappy, I've added a better "squeeze theorem" type thing for the derivative of $\sin(x)$. $\endgroup$ Jan 5, 2017 at 23:14
8
$\begingroup$

The answer ultimately depends on how you define $\sin x$ in the first place.

Here is a more fun one! $\sin x$ is the unique function satisfying $$ y'' = -y; y(0)=0, y'(0)=1 $$ By Theory of Ordinary Differential Equations a unique function defined on some interval containing $0$ exists. Now that we called it $\sin x$, we see that $$ \frac{\sin x}{x} = \frac{\sin x - \sin 0}{x - 0} \to y'(0) = 1 \ ,$$ by definition of derivative at $x=0$ and the initial conditions defining $y(x)=\sin x \ .$

$\endgroup$
1
  • $\begingroup$ I like this answer, though it would not have been very useful for me back then. $\endgroup$
    – FUZxxl
    Nov 21, 2019 at 0:31
7
$\begingroup$

Here is another approach.

enter image description here(1) picture 2(2)

In the large triangle, $$\tan(\theta)=\frac{opp}{adg}=\frac{z}{1}=z$$ So the triangle has height $$z=\tan(\theta)$$ and base $1$ so it's area is $$Area(big triangle)=\frac{1}{2}z=\frac{1}{2}\tan(\theta)$$

Next the sector area as a fraction of the entire circle, the sector is (see the right hand side of picture (1))$$\frac{\theta}{2\pi}$$ of the entire circle so it's area is

$$Area(sector)=\frac{\theta}{2\pi}*(\pi)(1)^2=\frac{\theta}{2}$$ The triangle within the sector has height $y$. But $y=\frac{y}{1}=\frac{opp}{hyp}=\sin(\theta)$ so the small triangle has height $y=\sin(\theta)$ and base $1$ so its area is $$Area(small triangle)=\frac{1}{2}y=\frac{1}{2}\sin(\theta)$$ Now we can use the sandwich theorem as $$Area(big triangle)\geq Area(sector)\geq Area(small triangle)$$

using the equations we worked out above this becomes

$$\frac{\tan(\theta)}{2}\geq\frac{\theta}{2}\geq\frac{\sin(\theta)}{2}$$ now multipliying through by two and using the fact that $$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$$ we get that $$\frac{\sin(\theta)}{\cos(\theta)}\geq\theta\geq\sin(\theta)$$ taking reciprocals changes the inequalities so we have that $$\frac{\cos(\theta)}{\sin(\theta)}\leq\frac{1}{\theta}\leq\frac{1}{\sin(\theta)}$$ now finally multiplying through by $\sin(\theta)$ we get $$\cos(\theta)\leq\frac{\sin(\theta)}{\theta}\leq1$$ Now $$\lim \limits_{\theta \to 0}\cos(\theta)=1$$ and$$\lim \limits_{\theta \to 0}1=1$$

so by the sandwhich theorem $$\lim \limits_{\theta \to 0}\frac{\sin(\theta)}{\theta}=1$$ also. QED

$\endgroup$
7
$\begingroup$

How about this proof?

We can check that function defined as \begin{align} \int_{-m}^m e^{2\pi i k x} \mathrm{d}{k} \end{align} is continuous and have a value $2m$ at $x=0$.

It is same with \begin{align} \int_{-m}^m e^{2\pi i k x} \mathrm{d}{k} = \dfrac{e^{2\pi i m x} - e^{-2\pi i m x}}{2\pi i x} = \dfrac{\sin 2 \pi m x}{\pi x} \end{align}

In conclusion, \begin{align} \lim_{x \to 0}\dfrac{\sin 2 \pi m x}{\pi x} = \lim_{x \to 0}\int_{-m}^m e^{2\pi i k x} \mathrm{d}{k} = 2m \end{align}

You can adjust $m$ as what you want.

$\endgroup$
7
$\begingroup$

Areas

This is a variant of robjohn's answer. The area of the sector $ADE$ is $\frac{1}{2}x\cos^2(x)$; the area of the triangle $ABC$ is $\frac{1}{2}\sin(x)$; and the area of the sector $ABC$ is $\frac{1}{2}x$. By inclusion, we find that for $0<x < \frac{\pi}{2}$, $$ \frac{1}{2}x\cos^2(x) \le \frac{1}{2}\sin(x) \le \frac{1}{2}x \, . $$ If we multiply each term in this inequality by $\frac{2}{x}$, we obtain the necessary bounds to apply the squeeze theorem: $$ \cos^2(x) \le \frac{\sin(x)}{x} \le 1 \, . $$ This inequality is also valid for $-\frac{\pi}{2} < x < 0$, as all three functions in the inequality are even.

Since $\cos$ is continuous at $0$, and $(\cdot)^2$ is continuous at $\cos(0)$, the function $x\mapsto \cos^2(x)$ is continuous at $0$, i.e. $$ \lim_{x \to 0}\cos^2(x)=1 \, . $$ Therefore, by the squeeze theorem, $$\lim_{x \to 0}\frac{\sin(x)}{x}=1 \, .$$

$\endgroup$
2
  • $\begingroup$ I like also this approach. It is similar to $\sin x <x < \tan x$. $\endgroup$
    – Sebastiano
    Aug 4, 2021 at 22:33
  • $\begingroup$ This is a late answer, but it is a good approach. $\endgroup$ Aug 4, 2021 at 23:58
7
$\begingroup$

For completeness answers let me suggest axiomatic approach to $\sin$ and $\cos$. One possible definition is here. I find another one(Ilyin, Poznyak: Fundamentals of Mathematical Analysis, 2005, vol.1, pages 146-155, Russian lang.), which claims, that there is only one pair of continuous functions on $\mathbb{R}$ for which fulfill

$$S(x+y)=S(x)C(y)+S(y)C(x)$$ $$C(x+y)=C(x)C(y)-S(y)C(x)$$ $$S^2(x)+C^2(x)=1$$ $$S(0)=0,S\left(\frac{\pi}{2}\right)=1,C(0)=1,C\left(\frac{\pi}{2}\right)=0$$ From this axioms can be obtained monotonic properties and can be proved, particularly, that for $x \in (0, \frac{\pi}{2})$ holds $0<S(x)<x< \frac{S(x)}{C(x)}$. Using last is easy to obtain $$\lim_{x \to 0}\frac{S(x)}{x}=1$$

$\endgroup$
4
  • $\begingroup$ Where is the definition of pi used? I'm asking because it seems the entirety of the proof is relegated to the last sentence with the inequalities 0<S(x)<x<S(x)/C(x), which only hold when you are using radians $\endgroup$
    – latbbltes
    Apr 25, 2021 at 18:49
  • 1
    $\begingroup$ Added source to answer. From proof can be seen, that in place of $\frac{\pi}{2}$, it can be taken any $d>0$. Connection with circle, then will be obtain from definition of $\pi$ based on circle en.wikipedia.org/wiki/Pi#Definition $\endgroup$
    – zkutch
    Nov 2, 2021 at 8:24
  • $\begingroup$ I'm not literally asking for the definition of pi. I'm asking how you derive the inequality since it doesn't follow simply from the formal properties stated without further work $\endgroup$
    – latbbltes
    Nov 2, 2021 at 20:22
  • $\begingroup$ How do I know if you are asking literally or not? You asked a question, I answered. The more precise your questions are, the more accurate my answers will be. You asked where pi is used - you can see it in the fourth line. You state that "it doesn't follow simply", but the word simple is subjective and depends on the person. In my comment, prior to your question, I included in the answer an indication of the source of the proof where the required properties are obtained on 9 pages. I leave it up to the reader to decide what is easy and what is difficult. $\endgroup$
    – zkutch
    Jan 24 at 23:39
6
$\begingroup$

This is a new post on an old saw because this is one of those things where that I can see how that, all too sadly, the way in which we've structured the current maths curriculum really doesn't make it possible to do these kinds of things the justice they deserve and I think, ultimately, that is a disservice to many learners.

The truth is, this limit cannot really be given an honest proof without an honest definition of the sine function, first. And that is not as easy as it seems. Even if we consider the simple notion from many trigonometric treatments that the sine is equal to the "length of the opposite side of the right triangle divided by the length of its hypotenuse", that doesn't truly solve the problem because there is actually a subtle missing element and that is that sine is not a function of a "right triangle" (though you could define that if you wanted, and it'd be easy!), but of an angle measure. And actually parsing out what "angle measure" means, it turns out, is essentially equivalent to defining the sine function in the first place, so this approach is circular! (pun observed after writing despite not being originally intended!)

So how do we define sine, or angle measure? Unfortunately, any approach to this is such that it must involve calculus. This is because the angle measure we use is "smooth and steady", meaning that, basically, if we have some angle, we'd like fractionating that angle measure to fractionate the angle in the same manner as cutting up pieces of a pie: if I have an angle with the given angle measure $\theta$, then for the measure system to work I should be able to then produce an angle with measure $\frac{\theta}{n}$, should be an angle that is geometrically the $n$-section of the angle into $n$ congruent smaller angles that add up to the full angle.

Yet already, we can see right there that this is not trivial: consider $n = 3$. Then we have the famous "impossible" problem of "trisection of the angle" which vexed even the ancient Greeks and for which people would keep trying to pound at until Pierre Wantzel finally proved it undoable over two thousand years after. We are asking for a mathematical widget that can not only trisect, but 5-sect, 629-sect, etc. angles and in a systematic manner to boot!

Indeed, not only is the sine function not trivial, we could argue that even the exponential function is considerably easier to treat than sine, though I won't give such a treatment here.

Thus, how do we do it? Well, the key observation is that our "steady" angle measure is one which is, effectively, defined by the arc length of a segment of circle intercepted by the angle when drawn at the circle's center and projected outward. In particular, this should be "obvious" from the (circularly-introduced) geometric formula

$$\mbox{Length of circular arc} = r\theta$$

Since this is only a trivial multiplication, all the nontriviality must be in either defining $\theta$ in terms of geometric angles formed by lines, or in terms of defining the "length of a circular arc" and, moreover, these two problems must be equally hard. Hence, we will begin with the arc question first and one will see that this answer will end up using a fair bit of Calculus II material to answer this Calculus I-level question about a supposedly pre-Calculus mathematical object. Indeed, this is what the whole "radian measure" is: it's a measure of angles in terms of the arc length of the piece they cut from a unit circle (i.e. $r = 1$). "Degrees", are then just a weird multiple unit of actual length, equal to $\frac{2\pi}{360}$ (or better, $\frac{\tau}{360}$) of some other unit length.

If you use a somewhat more honest Trigonometry book, you will see something to the effect that sine and cosine are defined as basically being the coordinates on a unit circle when an angle measure $\theta$ has been emplaced from the $x$-axis:

$$C(\theta) := (\cos(\theta), \sin(\theta))$$

Now as said above, $\theta$ is arc length. Thus, what we have above is something called an arc length parameterization of the circle - and that tells us how we need to proceed. First, we need a separate definition of the arc length of a circle.

How do we get that? Well, we will obviously need a more elementary circle equation, first, than the one we just gave, and that means going to the simple algebraic definition,

$$x^2 + y^2 = 1$$

so that now we can solve using good ole' Algebra for $x$ and $y$ in at least a semicircle:

$$x(y) = \sqrt{1 - y^2}$$ $$y(x) = \sqrt{1 - x^2}$$

And now this is where we then must introduce Calculus II-level concept - namely, integration for arc length. The arc length swept between two values of the $x$-coordinate, for a curve given with $y$ as a function of $x$, is

$$\mbox{Arc Length}(x_1, x_2) := \int_{x_1}^{x_2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$

Hence for the circle, now $\frac{dy}{dx} = \frac{1}{2} (1 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{1 - x^2}}$, so

$$\mbox{Arc Length}(x_1, x_2) = \int_{x_1}^{x_2} \sqrt{1 + \frac{x^2}{1 - x^2}} dx$$

which simplifies to

$$\mbox{Arc Length}(x_1, x_2) = \int_{x_1}^{x_2} \frac{1}{\sqrt{1 - x^2}} dx$$

Now, we use the Fundamental Theorem of Calculus to define the inverse sine as

$$\arcsin(x) := \int_{0}^{x} \frac{1}{\sqrt{1 - \xi^2}} d\xi$$

which is the arc length in terms of coordinate, and now the sine is its inverse, coordinate in terms of arc length:

$$\sin(\theta) := \arcsin^{-1}(\theta)$$

.

Finally, at this point, with a full, airtight definition of $\sin(x)$ now in hand, we are ready to evaluate the limit:

$$\lim_{x \rightarrow 0} \frac{\sin(x)}{x}$$

Since the "real", or base, function here is really the inverse function, i.e. $\arcsin$, we first proceed by making a change of variables: we consider instead the limit in terms of $y$ where $y(x) := \arcsin(x)$. Note that, trivially, $\arcsin(0) = 0$ from the integral definition, thus we get

$$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = \lim_{y \rightarrow 0} \frac{y}{\arcsin(y)}$$

Now for the right-hand limit, we need only consider the behavior of $\arcsin(y)$ when $y$ is small. Since the integrand, $\frac{1}{\sqrt{1 - x^2}}$, is differentiable at $x = 0$, it can be approximated with its tangent line (which really, should be also how we define tangent lines in the first place, as a "best approximation", a notion that can be done in an airtight, intuitive fashion through the use of a "zoom-in") and so likewise, the integral over a tiny sliver by integral of that same tangent line. By the power rule and chain rule,

$$\frac{d}{dx} \frac{1}{\sqrt{1 - x^2}} = \frac{d}{dx} (1 - x^2)^{-1/2} = \left(-\frac{1}{2}\right) (1 - x^2)^{-3/2} \cdot (-2x) = x(1 - x^2)^{3/2}$$

so the derivative at $x = 0$ is zero and the tangent line is horizontal: since also $\frac{1}{\sqrt{1 - x^2}}$ evaluated at $x = 0$ is $1$, the tangent is

$$T(x) := 1$$

hence

$$\int_{0}^{y} \frac{1}{\sqrt{1 - \xi^2}}\ d\xi \approx \int_{0}^{y} 1\ d\xi$$

when $y \approx 0$, and then the right-hand integral is approximately $y$, hence $\arcsin(y) \approx y$ when $y \approx 0$ and

$$\lim_{y \rightarrow 0} \frac{y}{\arcsin(y)} = \lim_{y \rightarrow 0} \frac{y}{y} = \lim_{y \rightarrow 0} 1 = 1$$

hence

$$\lim_{x \rightarrow 0} \frac{\sin(x)}{x} = 1$$

QED.

Nonetheless, as I mentioned before, this doesn't solve the requirements of the question which, while I'm sure its original asker has long moved on is, nonetheless, still relevant to calculus student after calculus student up to today: prove the limit using only Calculus I/pre-Calculus methods. What I am saying is that, in fact, that is not truly honestly possible and reveals a weakness of the curriculum in that it doesn't actually follow the proper logical buildup of the mathematical edifice.

What really should be done is to leave trig for later, that is, skip trig and go for Calculus first. When I studied maths on my own, I did just that. In fact, I'd say, as many educators have suggested, that most people don't need either, but really need more statistics instead. Then for those who do pursue higher maths, if we've done algebra and statistics, we already have right there a lot of interesting material we can build on for calculus, including the exponential function. There is no need to add trig functions to "sweeten the mix" when it's already plenty sweet with integrals of algebraic functions like $x \mapsto \sqrt{1 - x^2}$ which is a very nice example of the area-integration relation, and such can, if emphasized more heavily, potentially invite more nuanced thinking about integrals beside just "plugging and chugging integration rules". In particular, with a more limited set of functions, we can think about other ways we might approach them like just that and/or a variety of ways to interpret the integral which can only be good, I'd think, to develop more creative thinking about problems and less drilling in methods with little real understanding gained (and rote crunching integrals is even less relevant now with computer algebra software; more important is really being able to understand a problem and how its parts fit together and lead to a solution. That said, rote crunching is not something I suggest banning either but I suggest that ideas, concepts, and creativity should come first, then you get into those techniques because very often they are also still useful in analysis and being fluent at them can also make you able to solve problems more quickly, e.g. you don't want to be hitting up your calculator for 2+3 all the time in grade school and you don't want to be hitting up your CAS all the time to integrate $x \mapsto x^2$). On top of that, we have effectively now two different functions - logarithm and trigonometry - which we define by integrals, which means also that we can consider that we don't have to stop there, and this exposes the artificiality of the sacredness of so-called "elementary functions" and alows us to also perhaps consider a few more artifices of that sort like $\mathrm{erf}(x)$ which seems not too much more difficult but instead we simply reply that

$$\int e^{-x^2}\ dx$$

"can't be done" which, in light of having seen such things early on, feels like another sore cheat/blemish on the curriculum.

And to finish it all off - if you say calc can't be done before trig, I'd say that too bad Archimedes isn't here, as he would probably not have shared your sentiment since in fact he was one of the earliest to develop even a partial concept of integration and not only that but one of his applications of it was precisely the delineation of the arc length of a circle: that is why $\pi$ is called Archimedes' constant.

$\endgroup$
9
  • $\begingroup$ If the (real) exponential function is easier to define than the trigonometric functions, then why not go by way of the imaginary exponential? And BTW, $\theta$ is not arc length, except for the unit circle. It is only proportional to arc length. An intuitive way to think about theta is to think rotations. $\endgroup$
    – Allawonder
    Jun 24, 2019 at 3:38
  • $\begingroup$ There are many other little "errors" -- shall we say -- to respond to here, but well, to do some calculus richly well you need some nonalgebraic functions -- the exponential, both real and imaginary, suffices, with its inverse. About why we call these elementary. Well, similar to what you noted -- it's somewhat easy to introduce the exponential function by analogy with addition and multiplication -- at least initially. Thus, one can learn much of these functions without any calculus. But note that even all the algebraic functions are not rigorously understood yet without calculus -- in... $\endgroup$
    – Allawonder
    Jun 24, 2019 at 3:51
  • 1
    $\begingroup$ @Allawonder : I did mention that the radian measure specifically is defined with a unit circle. My point was that measuring angles, in the way we do, is equivalent to measuring the arc length of a circle . That said, I also realized after writing this that a simpler method could be to consider repeated bisection of an angle, but you still need a limit process to fill in at all non-dyadic points, and this can introduce degree measure starting with the definition that a right angle is to be measured as 90 degrees. $\endgroup$ Jun 24, 2019 at 5:38
  • 1
    $\begingroup$ Also, reagrding the real line - yes, it is true that technically even elementary algebra on $\mathbb{R}$ requires "calculus" in this sense because that the real number line itself could be considered to be the essence of calculus, and in fact that is what motivates its definition. However, I'd think the intuitive picture of the line as a "line" is actually fairly sufficient for elementary-algebraic needs without getting into the nitty-gritty of the microstructure. The biggest nit for me there is the use of infinite decimals - which leads many people to the $\endgroup$ Jun 24, 2019 at 5:43
  • 1
    $\begingroup$ persistent misconception of $0.9999... \ne 1$, something that can't be truly disabused honestly without a closer look. My solution to that is just don't bother with infinite decimals. Stick with finite decimals, and say that most reals can only be approximated. $\endgroup$ Jun 24, 2019 at 5:43
5
$\begingroup$

We can also use Euler's formula to prove the limit:

$$e^{ix} = \cos x + i\sin x$$

$$\lim_{x\to 0}\dfrac{\sin x}{x} = \implies \lim_{x\to 0} \dfrac{e^{ix}- e^{-ix}}{2i x}$$

$$= \lim_{x\to 0} \dfrac{e^{2ix}-1}{2ix}\times\dfrac 1{ e^{ix} }= 1 \times 1 = 1$$

since:

$\lim_{f(x)\to 0}\dfrac{e^{f(x)}-1}{f(x)} = 1$

$\endgroup$
1
  • 6
    $\begingroup$ If in the end you are going to use some "known" limit, why not doing that in the beginning? $\endgroup$ Apr 29, 2019 at 14:16
4
$\begingroup$

Here is a proof to those familiar with power series.

The definition of $\sin(x)$ is

$$\sin(x) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}h^{2k+1}$$

Therefore we get

$$\begin{align} \lim_{x \to 0} \frac{\sin(x)}{x} &= \lim_{x \to 0} \frac{\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!}x^{2k+1}}{x} \\&= \lim_{x \to 0} \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k} \\&= 1 + \lim_{x \to 0} \sum_{k=1}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k} \\&= 1 \end{align}$$

where we have used the fact that the power series $\sum_{k=1}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k}$ has radius of convergence $R=\infty$ and therefore is continuous on $\mathbb R$. This allows us to take the limit inside and we get

$$\lim_{x \to 0} \sum_{k=1}^\infty \frac{(-1)^k}{(2k+1)!} x^{2k} = \sum_{k=1}^\infty \frac{(-1)^k}{(2k+1)!} 0^{2k} = 0$$

$\endgroup$
4
  • 1
    $\begingroup$ Compare to this answer and this answer, which both use power series. $\endgroup$
    – robjohn
    Jan 14, 2018 at 19:48
  • 3
    $\begingroup$ The premise of the question is not to use power series. $\endgroup$
    – FUZxxl
    Dec 4, 2018 at 12:34
  • $\begingroup$ @FuZxxl That depends how you define "sine". The definition such that sine is the ratio of opposite side to the hypotenuse is very vague while using as a proof. Therefore, you can choose to define sine as what philmcole suggested, with illustration of this meaning that sin x is the ratio, then you can find the limit with power rule. $\endgroup$ Sep 20, 2021 at 3:00
  • $\begingroup$ @KelvinChan You can, but as I already explained in a bunch of other comments, the definition based on ratio between opposing cathetus and hypothenuse is what is meant to be used. $\endgroup$
    – FUZxxl
    Sep 20, 2021 at 8:49
1
$\begingroup$

This is not a rigorous proof, but is instead an intuitive argument. Consider the graph of the sine function and in particular consider the origin $(0,0)$ and some arbitrary point $(x,\sin(x))$ a little bit to the right of it. Connect a secant line between the two points like so:enter image description here Now consider the average rate of change of the sine function on that interval $[0,x]$, or alternatively the slope $m$ of that secant line, namely: $$m(x)=\frac{\sin(x)-\sin(0)}{x-0}$$ But clearly this is just $$m(x)=\frac{\sin(x)}{x}$$ But then "clearly" (again, this is just for intuitive purposes), as $x\to{0}$, the slope of this secant line approaches the slope of the tangent line to the graph at $x=0$ (i.e. it approaches whatever $\sin'(0)$ should be). Looking at the graph then, it shouldn't feel too unreasonable that this limiting procedure results in a $45$-degree diagonal tangent, hence a slope of $1$.

$\endgroup$
1
  • 2
    $\begingroup$ This has already been said before. $\endgroup$
    – Toby Mak
    May 21, 2021 at 0:00
1
$\begingroup$

Here is a slick trick using elementary integration methods. Note that \begin{align} \int_0^1 \ \cos(xt) \ dt & = \left[ \dfrac{1}{x} \cdot\sin(xt) \right]_0^1 \\ & =\dfrac{\sin (x)}{x} - \dfrac{\sin (0)}{x} \\ & = \dfrac{\sin (x)}{x}. \end{align} Hence, \begin{align} \lim_{x \rightarrow 0} \dfrac{\sin (x)}{x} & = \lim_{x \rightarrow 0} \int _0^1 \ \cos(xt) \ dt \\ & = \int_0^1 \cos (0) \ dt \\ & =1. \end{align}

$\endgroup$
1
  • 2
    $\begingroup$ Is there a way to know the antiderivative of $\cos$ without knowing the limit in question? I mean, the limit in question is usually used to prove that $\sin' =\cos$. $\endgroup$
    – Gary
    Jan 1 at 7:40

Not the answer you're looking for? Browse other questions tagged or ask your own question.