The Squeeze Law

A friend has told you "some really cool rocks are about a mile down the ravine over there" and, having the afternoon free, you decide to do some exploring. The ravine is quite wide at its mouth and a pleasant brook runs down its floor.

You start wandering down the ravine and after a while it strikes you that the sides of the ravine have moved much closer together and the brook has grown considerably in volume. You keep going, but just a few minutes later you stop. Just ahead the ravine narrows to the point where there is no floor left other than that occupied by a fairly gnarly looking stream, and if you keep going you will be literally stuck between a rock and a hard place. Oh well, it was a nice afternoon anyway, so you turn around . . .

In the vignette above, you were the victim of the squeeze law. The squeeze law says that if the values (or, equivalently, the graph) of a function g(x) is "squeezed" in between the values of two functions f(x) and h(x), and if at some point a both f and h have the same finite limiting value L, then g(x) also approaches L as x approaches a. Stated more formally:

The Squeeze Law
Let f(x), g(x), and h(x) be defined at every point (with the possible exception of a) of an open interval J containing a. Suppose that for every point x in J we have that
f(x) g(x) h(x)
and that
lim f(x) = L and lim h(x) = L.

x --> a x --> a

Then lim g(x) = L. (In particular, the limit exists.)

x --> a

Why is THAT true?

Example:
Graph of w = z*sin(pi/z) (red), w=z & w=-z (green).
We have seen that Pork Chop's function p(z) = sin(/z) is very badly behaved and oscillates wildly as z nears zero. The limit of p(z) as z approaches zero does not exist. But now consider a variation on this function:

q(z) = z sin(

/z).

Multiplying by z has the effect of putting an envelope on the function p(z).

To apply the squeeze law, we note that |sin(/z)| is never larger than 1, so

|q(z)| = |z sin(

/z)| = |z| |sin(

/z)|

|z|.

It follows that, for all real z except z=0,

-z

q(z)

But

lim	-z = 0 and	lim	z = 0.
z --> 0		z --> 0

Hence by the squeeze law we conclude that

lim	z sin(/z) = 0.
z --> 0

APPLICATION

There is one extremely important application of the squeeze law, and that is the proof that

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

To work through this proof, which is based on some simple but pretty geometry, take the side trail that starts here.

/Stage4/Lesson/squeeze.html

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