Show that $\lim_{x \to 0} \frac{\sin\, x}{x} = 1$

To motivate the limit computation of this section, using Sage we compute a number of values of the function $\frac{\sin\, x}{x}$, as $x$ gets closer and closer to 0:

$x$	0.5000	0.2500	0.1250	0.06250	0.03125
$\frac{\sin(x)}{x}$	0.9589	0.9896	0.9974	0.9994	0.9998

Indeed, if we refer to the table in §1.4, it will be seen that for all angles less than $10^o$ the angle in radians and the sine of that angle are equal to three decimal places. To compute the table of values above using Sage, simply use the following commands.

[fontsize=\small,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: f = lambda x: sin(x)/x
sage: R = RealField(15)
sage: L = [1/2^i for i in range(1,6)]; L
[1/2, 1/4, 1/8, 1/16, 1/32]
sage: [R(x) for x in L]
[0.5000, 0.2500, 0.1250, 0.06250, 0.03125]
sage: [R(f(x)) for x in L]
[0.9589, 0.9896, 0.9974, 0.9994, 0.9998]

From this we may well suspect that $\lim_{x \to 0} \frac{\sin\, x}{x} = 1$.

Let $O$ be the center of a circle whose radius is unity.

Let $\operatorname{arc}\ AM = \operatorname{arc}\ AM' = x$, and let $MT$ and $M'T$ be tangents drawn to the circle at $M$ and $M'$. From Geometry (see Figure 3.12),

Figure 3.12: Comparing $x$ and $\sin(x)$ on the unit circle.

we have

$$MPM' < MAM' < MTM';$$

or $2 \sin\, x < 2x < 2 \tan\ x$. Dividing through by $2 \sin\, x$, we get

$$1 < \frac{x}{\sin\, x} < \frac{1}{\cos\ x}.$$

If now $x$ approaches the limit zero,

$$\lim_{x \to 0} \frac{x}{\sin\, x}$$

must lie between the constant $1$ and $\lim_{x \to 0} \frac{1}{\cos\ x}$, which is also $1$. Therefore $\lim_{x \to 0} \frac{x}{\sin\, x} = 1$, or, $\lim_{x \to 0} \frac{\sin\, x}{x} = 1$ Theorem 3.8.3. $\qedsymbol$

It is interesting to note the behavior of this function from its graph, the locus of equation

$$y = \frac{\sin\, x}{x}$$

Figure: The function $\frac{\sin(x)}{x}$.

Although the function is not defined for $x = 0$, yet it is not discontinuous when $x = 0$ if we define $\frac{\sin\, 0}{0} = 1$ (see Case II in §3.6).

Finally, we show how to use the Sage command limit to compute the limit above^3.5.

[fontsize=\small,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: limit(sin(x)/x,x=0)
1