The sine function divided by its angle, and a certain limit

What is $ \lim_{x\rightarrow0} \frac{\sin x}{x}$ ? Recall the definition of a limit, repeated here for reference [2].

A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x-a| < \delta $ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it). In symbols we write $\lim_{x \rightarrow a} f(x) = A$.

Using this definition, $a$ is $0$ in this case. Now I extract as much information from the limit definition as possible, for this case of $a$ of $0$. Using the language from reference [2], and replacing $a$ with $0$, one has:

A function $f(x)$ approaches a limit $A$ as $x$ approaches $0$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x| < \delta $ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $0$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it). In symbols we write $\lim_{x \rightarrow 0} f(x) = A$.

Using the previous post about the definition of a limit, it is possible to obtain information about $x$ and $f(x)$ for the case of $a=0$. It is helpful to draw the situation.

This analysis applies for each positive number that $\delta$ can be, which implies that $\delta$ can be very close to zero without being zero.

To find the limit, one must find the value of $A$. How should this be done? In this case, $f(x)=\frac{\sin x}{x}$. Perhaps it possible to view this situation graphically, by using the values of sine with angles in radians. Here, the angle is $x$.

Start with the unit circle.

Next recall the definition of a radian. Using a unit circle, one radian corresponds to one unit of length since the radius of a unit circle is one unit of length. Using the result that the circumference of a circle with radius $R$ is $2\pi R$ units of length, the circumference of the unit circle is $2\pi$ units of length. Note that $\pi$ is essentially a scaling factor so that the circumference of a circle is some multiple of the circle’s diameter, $2R$. Using the concept of proportions,

$ \displaystyle \frac{1 [rad]}{1 [u]} = \frac{\alpha [rad]}{2\pi [u]}$.

Here, $\alpha$ is the number of radians corresponding to the circumference of the unit circle, and $u = [u]$ is one unit of length. Also, $[rad]=rad$. The brackets are used for aesthetics. To satisfy this equality, the value of $\alpha$ must be $2\pi$. Thus the number of radians corresponding to a circumference of $2\pi$ units of length is $2\pi$.

For the case of $ \lim_{x\rightarrow0} \frac{\sin x}{x}$, the definition of a limit requires the angle $x$ to have positive and negative values. One can define negative and positive angles, and there is no geometrically-obvious definition. By convention, positive angles are those corresponding to counter-clockwise motion about a circle, and negative angles correspond to clockwise motion about a circle. This convention could be flipped so that negative angles correspond to counterclockwise motion but mathematicians do not usually use this convention; it is questionable whether one convention is better than the other. In both cases, the angle of zero radians distinguishes the positive angles from the negative angles. Also, note that the idea of counterclockwise motion depends on whether the observer is above or below the circle.

This ambiguity can also be applied to a number line. An observer looking at a number line located in front of the observer claims, following the usual convention, that positive numbers exist to the right of zero while negative numbers exist to the left of zero. But another observer on the other side of this number line–that is, behind the number line, has a different story with the same convention. This observer notices that the positive numbers exist to the right of zero and negative numbers exist to the left of zero, but left and right are different for each observer. The first observer notices positive numbers on the right of zero whereas the second observer behind the number line notices positive numbers on the first observer’s left and the second observer’s right. The convention of positive numbers to the right of zero and negative numbers to the left of zero is not broken, but the two observers disagree on the locations of the positive and negative numbers.

Theoretically, it is possible to break this kind of convention and declare, arbitrarily, that positive numbers are to the right of zero for an observer looking ahead at a number line, while positive numbers are to the left of zero for an observer behind the number line as seen from the first observer. Then there is no ambiguity as to where the positive numbers are on the same number line.

This type of convention can also be broken for the positive and negative angles which correspond to displacements or signed differences along the circumference of a circle. One can declare, arbitrarily, that the positive angles correspond to counterclockwise rotations with respect to an observer looking down at a unit circle whereas an observer looking up at the same unit circle associates clockwise motion with positive angles and counterclockwise motion with negative angles. So, the positive-negative conventions are reversed for opposite observers, but the values of the angles on the unit circle remain the same regardless of the observer, which is logical in a sense.

In light of the above discussion, an observer above the unit circle notices positive $x$-values to the right of zero for the $x$-axis whereas an observer below the unit circle notices positive $x$-values to the left of zero for the $x$-axis.

No matter how the positive and negative angles and $x$-values are laid out, the main point is that it is possible to define and use positive and negative values for angles and positions.

Evaluating $ \lim_{x\rightarrow0} \frac{\sin x}{x}$ involves positive and negative angles. The angles which are positive are close to zero and the angles which are negative are close to zero. The angle $x$ can be any number except for zero which, for instance, allows for very small positive numbers. These very small positive numbers are the absolute values of the corresponding negative numbers that are close to zero.

Should the value of the sine function depend on whether the angle is positive or negative? Geometrically, using $\sin x = \frac{opp}{hyp}$, it is impossible for the sine function to be negative since lengths are positive by definition, but from experience I know that this is not the popular convention for the sine function. So it might be necessary to change this geometrical definition of the sine function to match modern conventions, most likely based on the unit circle.

The above explanation does not explicitly address negative angles, but it is straightforward to modify the above argument to include them, due to symmetry about the $x$-axis.

Generally, in evaluating the limit of an expression it is useful to understand the trend(s) of the expression near $x=a$, and knowing information about the expression and/or its geometrical parts at $x=a$ is helpful as well. Note that the expression does not need to be defined at $x=a$ for the limit to be defined at $x=a$–this is a consequence of the wording in the limit definition. Note also that there is no functional relation between $\epsilon$ and $\delta$ [1]. What is the case is that $\epsilon$ can be a very small number.

References

[1] https://math.stackexchange.com/questions/3968287/when-does-a-function-not-have-a-limit

[2] David V. Widder. Advanced Calculus. Dover 1989.

[3] https://math.stackexchange.com/questions/3967118/limit-of-a-function-with-an-undefined-feature

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