# Algebraically compute $\lim_{x \to 0}\frac{\sin x}{x}$

I know that: $$\lim_{x \to 0}\frac{\sin x}{x} = 1$$ However all proofs I have seen resort to some kind of geometric appeal or otherwise fall to L'Hôpital's rule or Taylor series. However, I want an to see an proof that is purely algebraic (I mean least geometric intuition, mainly not diagrams to convince the reader of something, but obvious algebraic statements one after another) that uses nothing advanced like series and [I mean accessible to ordinary calculus beginner]... I do not want want elipson delta proofs but something that has quite evidently, no gaps.

Edit: Some comments have said sine cannot be defined without geometry or series. Well, I meant to say that we can assume that we know the unit circle definition. I only meant to say that we do not rely on diagrams in the proofs. We know the obvious facts that what sine and radians are, etc. I did not mean it to take to formally and rigorously.

Hope you guys have some elegant and delightful proof of the kind. It is something which is bothering me for quite a while.

• Please define $\sin x$. – Hagen von Eitzen Jan 5 '14 at 17:42
• How do you define the sine function? – Andrés E. Caicedo Jan 5 '14 at 17:43
• The "unit circle" definition presumably says something like "at angle $\theta$, draw a ray; the point where it intersects the circle is $(\cos \theta, \sin \theta)$. Would you be willing to modify that to "From the point $(1, 0)$, go counterclockwise around the circle for a distance of $\theta$; the point you find has coordinates $(\cos \theta, \sin \theta)$"? Or do you have a good definition for "angle," too? And are you willing to accept that for small $x$, $|\sin x| < |x|$? And what about the addition formulas for sine and cosine -- are they allowed? – John Hughes Jan 5 '14 at 17:55
• It looks like you want a proof without geometry, series. l'Hospital's Rule...without mathematics?! And you want something "algebraic" but "not advanced". I don't think there's such a thing, and the geometric proof is so simple that even high school students can understand it. – DonAntonio Jan 5 '14 at 17:58
• @alexjo No that is not a duplicate because the question actually asks for geometric appeal where it is the opposite here. – Sawarnik Jan 5 '14 at 18:16

I believe the difficulty many students have in understanding the proof of $$\lim_{x\to0}\frac{\sin x}{x}=1$$ is that it often comes too early in a calculus course, before one has learned integration, and therefore before one has learned how to think about arc lengths and areas. Since arc length (or area) enters fundamentally into the definition of the $\sin$ function, the proofs that are usually given end up relying on geometric arguments that seem out of place next to the style of argumentation that one otherwise encounters in the first part of a calculus course. I'm not suggesting that these arguments aren't rigorous, or that there aren't good reasons for bringing up trigonometric functions early—but I believe it can be satisfying to understand how this result comes about, rather straightforwardly, once one has defined the trigonometric functions in calculus terms.

If definite integration and arc length are unfamiliar to you, then this proof may not help. However, the notion of arc length is used only to motivate a definition, and all one needs to know about the definite integral is that it represents the area under the curve. This is used to obtain upper and lower bounds.

We make a purely algebraic definition of the $\arcsin$ function as follows. For $y\in(-1,1)$ let $$\arcsin y=\int_0^y\frac{dt}{\sqrt{1-t^2}}.$$ Interpreted geometrically, this is, in fact, the formula for the signed length of the arc of the unit circle that extends from $(1,0)$ to $(\sqrt{1-y^2},y)$, and therefore indeed the $\arcsin$ function as usually defined, but this definition itself involves no geometry. The $\sin$ function is then the inverse of this function, at least in an interval containing 0. (In fact, in the interval $(-\pi/2,\pi/2),$ where the symbol $\pi$ represents twice the upper bound of the $\arcsin$ function. Thinking about how to extend the function to the entire real line is not needed for our purpose.)

We will first show that $$\lim_{y\to0}\frac{\arcsin y}{y}=1.$$ Since $\sqrt{1-t^2}$ is bounded above by $1,$ the integrand defining $\arcsin y$ is bounded below by $1,$ and, for $y$ positive, $\arcsin y$ is bounded below by $y.$ Hence $(\arcsin y)/y$ is bounded below by $1.$ Since, in the integral, $\sqrt{1-t^2}$ is bounded below by $\sqrt{1-y^2},$ the integrand is bounded above by $1/\sqrt{1-y^2},$ and, for $y$ positive, $\arcsin y$ is bounded above by $y/\sqrt{1-y^2}.$ Hence $(\arcsin y)/y$ is bounded above by $1/\sqrt{1-y^2}.$ Since $(\arcsin y)/y$ is an even function, the same bounds apply for $y$ negative. But both of these bounds have limit $1$ as $y\to0,$ and the result follows from the squeeze theorem.

This is almost the result we want. To finish, define the continuous function $$f(y)=\begin{cases}\frac{\arcsin y}{y} & \text{for y\in(-1,1)\setminus\{0\},}\\ 1 & \text{for y=0,}\end{cases}$$ by plugging the hole at $y=0$ using the limit we just computed. Let $g(x)=\sin x.$ Since $f$ and $g$ are continuous at $0$ and $g(0)=0,$ we have $$\lim_{x\to0}f(g(x))=f\left(\lim_{x\to0} g(x)\right)=f(0)=1.$$ But for $x\ne0,$ we have $$f(g(x))=\frac{\arcsin\sin x}{\sin x}=\frac{x}{\sin x},$$ and so we have our result.

Added remark: Continuity of arcsin on (−1,1) follows in the definition I have used from continuity of the integrand and the Fundamental Theorem of Calculus (which actually implies differentiability). Continuity of sin follows from the easy fact that arcsin is one-to-one on (−1,1) and from the theorem that the inverse of a continuous, one-to-one function is continuous. We have $\sin x= AX$ and $\tan x=UT$ and $x=UX$ and geometrically we have $$AX<UX<UT$$ hence dividing by $\sin x$ we obtain: $$1<\frac{x}{\sin x}<\frac{1}{\cos x}$$ finaly pass to the limit $x\to0$ to find the desired result.

Edit To clear up the idea we have $$UX=x\times\text{the radius of the unit circle}=x$$

and the lenght of the arc $UX$ is greater than the hypothenus of the triangle $AUX$ which's greater than $AX$ so $AX<UX$ and finaly the area of the triangle $OUT$ is greater than the area of $OUX$ wich gives $UX<UT$.

• $x \neq UX$ for any $x > 0$ – AlexR Jan 5 '14 at 18:04
• To make $AX<UX<UT$ rigorous is a bit of a sub-challenge, after all $UX$ is a curve (hopefully a rectifyable one). – Hagen von Eitzen Jan 5 '14 at 18:04
• $x=UX$ requires you to measure in radians which requires another geometric fact, that arc length is the angle, which may or may not be acceptable! Since OP does not specify what can and can't be used, I am afraid that any answer may be acceptable or any answer may be unacceptable. – user44197 Jan 5 '14 at 18:05
• to make this rigourous one usually use the area of OAX, OUT (triangles) and the area of the circular sector OUX instead of the lengths. it gives the same inequalities. This proves $\lim\limits_{x\to 0^+} \dfrac{\sin(x)}{x}$, and you can get $x\to 0^-$ by using the parity of $x \mapsto \dfrac{\sin(x)}{x}$ – imj Jan 5 '14 at 18:15
• But this is a (the) geometric proof the OP requested to be avoided... – DonAntonio Jan 5 '14 at 19:10

This may be more for fun than proof as all proofs require some notion of sine that is either a series or based on geometry.

You can start with the fact that $\sin(30^\circ) = 1/2$ and the half angle formula to calculate $\sin(15^\circ)$ and then $\sin(7.5^\circ)$ to get $$\sin(7.5^\circ) \approx .13053$$ So $\sin(x)/x \approx 0.017403$.

In fact in the limit you will get $$\sin(x)/x \rightarrow 0.017453$$ So the limit is $1$ only if you measure in radians and you need some geometry to define a radian.

By the way, the limit of ratio of $\sin(x)/x$ in degrees is $\pi/180$, so you now have a simple way to calculate $\pi$!

You might use the fundamental relation of trigonometry and the known limit $\lim_{x \rightarrow 0}\frac{1- \cos x}{x^2}= \frac{1}{2}$

$$\lim_{x \rightarrow 0} \frac{\sin x}{x}= \lim_{x \rightarrow 0} \frac{\sqrt{1-\cos^2 x}}{x}= \lim_{x \rightarrow 0} \sqrt{\frac{1-\cos^2x}{x^2}}= \lim_{x \rightarrow 0} \sqrt{\frac{(1-\cos x )}{x^2}(1+ \cos x)}= 1$$

• How do we come upon the known limit? – Sawarnik Jan 5 '14 at 18:41
• using Taylor series that you certainly know. I don't see any other proof – Matheman Jan 5 '14 at 18:51
• ok :) I do not see anyway out too. – Sawarnik Jan 5 '14 at 18:54
• This is what I meant in my comment above: the "known limit" requires either geometry or slightly more advanced tools, and we're into a vicious circle... – DonAntonio Jan 5 '14 at 19:14
• Yeah, this is a nonproof. – JLA Aug 9 '14 at 21:05