Estimation and Limits -- Some Limits Involving Sines and Cosines

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

These limits turn out to be quite important and we will need to know them later. Investigating these limits also gives us some practice with the limit concept.

The source of a mathematical problem often gives us clues about its answer. The second limit

Comes up when we are trying to determine the slope of the tangent to the curve y = cos x at the point x = 0 -- that is, the point (0, 1) on the curve.

We estimate the slope of the tangent by looking at a line through the point (0, 1) and a point (x, cos x) where x is close to 0. The slope of this line is

So the exact slope of the tangent is

the limit we want.

using these same ideas. Be as complete and thorough as you can.

Answer.

By now you have a fair amount of evidence that seems to indicate that

and this brings up one of the most important questions in mathematics and science -- How do you know when something is true? The ultimate answer to this question is that you truly know something is true only when you understand why it is true. In mathematics and the sciences there are several important ways in which we come to believe that something is true.

• Evidence: This is perhaps the most important route to understanding and it is certainly the first. For the two limits we are examining in this module we have already examined a great deal of evidence. This is, generally, a good first step in looking at limits like

    Lim   f(x)
  x --> a
  

  That is, we calculate f(x) for many different values of x close to a. But evidence can be misleading both in mathematics and in science more generally.

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  Use your CAS window to investigate the limit

    Lim   f(x)
  x --> 0
  

  where

  

  by calculating f(0.1), f(0.01), f(0.001), f(-0.01), f(-0.01), f(-0.001). What does this evidence seem to indicate? Do you agree with the conclusion suggested by this evidence? Try some other small numbers.

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  The example in the CAS work above illustrates a very common problem in mathematics and especially in science. We may be tempted to draw general conclusions from data that is too specific. For example, medical reseacrhers must be very careful to look at many different kinds of patients before concluding that a particular treatment is safe -- the treatment may be perfectly safe for most patients but might be life threatening for others.

• General understanding: For example, in medicine treatments that use substances that are already present in the human body are likely to be safe.

• Proof: Mathematicians have an additional tool -- we can often prove that something is true. This is in many ways the best way of being certain that something really is true. Ideally a proof not only shows that something must be true but it also gives us some insight into why it is true. We often have a choice of different possible proofs, some of which may be more illuminating than others.

Consider the figure below

Look at the yellow "pie-shaped" portion of a circle. The center of the circle is at the origin and the angle at the origin is x radians. The radius is cos x.

• Because the circle has radius cos x and the angle is measured in radians the area of the yellow piece is

  x (cos x^2) / 2.

• The area of the triangle whose vertices are at (0, 0), (cos x, 0), and (0, sin x) is

  (sin x) (cos x) / 2.

It is clear from the picture that

  lim   cos x = 1
x --> 0

sin x <= x

Thus,

which completes the proof.