Chapter 1: Limits.

Section 1.3 and 1.4 (More Complicated Limits and Continuity):

These last topics that we'll cover from now in Chapter 1 involve what's called "the topology of the real line," as well as looking a bit at trigonometry.

One very important tool in evaluating limits is an apparently obvious fact, known as the "Squeeze Theorem."

Suppose we have three functions, f₁, f₂, and f₃, and some deleted neighborhood of a point, "c,", call it N = (c - h, c) together with (c, c + h).  Now, suppose that, in this neighborhood, f₂ is trapped between f₁ and f₃ . (f₁ <= f₂ <= f₃ ) on N.  Finally, suppose that f₁ and f₃ are squeezing together near c:  In the language of limits, lim_x->c f₁ = L = lim_x->c f₃.  Then we can conclude, not surprisingly, since f₂ is trapped between them, that lim_x->c f₂ = L.

Here's a picture, using f₁ = - abs(x) and f₃ = abs(x) and f₂ = x*sin(1/x), and looking at what happens near c = 0:

So, even though f₂ is undefined at 0, we can easily see the limit.

The Squeeze Theorem is used frequently in more advanced calculus classes, but for us, there's one main use, and that's to investigate a very important limit:

lim_x->0 sin(x)/x.  Notice that this is of the form: 0/0, so it's "indeterminate," but there's no good way to manipulate it to cancel out the divide by zero problem.  But, if we look at the graph:

it appears that this function approaches 1, as x gets close to 0.  In fact, we're going to trap this function between the cosine and the secant functions, near 0:

But to prove that what we're seeing in the picture, above, is actually true, we'll need a bit of trigonometry.

For review, please see the following pages:

From PreCalculus:

Section 0609

Section 0610

We'll begin by looking at the applet, below, and noticing that, using RADIAN MEASURE, the area of a circular wedge is particularly easy:

A = (Theta/2)*radius -- as long as Theta is measured in RADIANS.

Then, we see that the wedge is trapped between a small area and a large area:

Applet:

If we divide the inequality in bold by 0.5*sin(Theta), we get that:

cos(Theta) < Theta/sin(Theta) < sec(Theta).  Taking the reciprocals, we have that:

sec(Theta)>sin(Theta)/Theta>cos(Theta).

Now, since lim_x->0 cos(Theta) = lim_x->0 sec(Theta) = 1 (no divide by zero problems with either of these), the Squeeze Theorem tells us that lim_x->0 sin(Theta)/Theta = 1.

(Here it is in the notes from last year:)

The function sin(x)/x is sometimes denoted by Si(x), other times by sinc(x), with Si(x) reserved for the indefinite integral of the sinc function.  Here is Mathworld's article on the Sinc(x) function, which finds applications all over engineering, even outside of calculus.

We'll come back to this when we start looking at the derivatives of the sine and cosine functions.  For now, just remember that everything we did ONLY WORKS in RADIAN MEASURE.  That's why, when we apply Calculus to Trigonometry, we'll always need to convert everything to RADIANS.

Next, Continuity: