The Limit of sin(x)/x As x Gets Small

Our final example in this section is
In Calculus, angles are measured in radians. If x is not measured in radians, this limit will not be 1.

	sin(x)
lim
x --> 0	x

Here is a table of values and a graph which pretty well indicate that the value of this limit is 1.

**Table of Values of sin(x)/x**
x (radians)	-.7	-.2	-.05	0	.01	.03	.3	1.4
sin(x)/x	.92031	.993347	.999583	***	.999983	.99985	.98506	.703893

Graph of sin(x)/x

If we know the answer, namely that

	sin(x)
lim		= 1,
x --> 0	x

what then is the question? The question we ask this time is

What Does This Limit Tell Us?

This limit says that as x gets small the ratio of sin(x) to x approaches 1. If the ratio of two numbers is close to 1, then the two numbers are about equal. Thus

for small values of x, sin(x) is approximately equal to x.

A good way to get a feel for this approximation is to try out a few numbers:

Enter a value for x		sin(x)	Number of decimal places of agreement

The approximation "x sin(x) for small values of x" is a useful estimation tool.

Satisfying Your Guides

Your guides, at the very least, seem to have a healthy distrust of evidence gathered from tables and graphs. Maybe they are just old and crotchety, but maybe they are once burned and twice shy. No matter. It is worth considering how one might go about establishing in other ways that

	sin(x)
lim		= 1
x --> 0	x

Amazingly, we will be able to show why this limit is correct (on an enrichment page) in the very next Stage.

/Stage3/Lesson/sinExample.html

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