LimitTrig.html

Limits for Trigonometric Functions

> restart;
with( plots ):

Warning, the name changecoords has been redefined

>

Auxiliary Procedures -- do not display

>    figure := proc( theta )
  local A, C, P, S, T, Title, t;
  C := implicitplot( x^2+y^2=1, x=-1..1, y=-1..1, tickmarks=[0,0] ):
  S := theta -> plot( [[0,0],[cos(theta),sin(theta)],[cos(theta),0]], color=blue ):
  A := theta -> plot( [cos(theta)*cos(t),cos(theta)*sin(t),t=0..theta], color=green ):
  P := theta -> pointplot( [ [1,0],
                           [cos(theta),0],
                           [cos(theta)*cos(theta),cos(theta)*sin(theta)],
                           [cos(theta),sin(theta)] ],
                           symbol=circle, symbolsize=12 ):
  T := theta -> textplot( {
                  [-0.1,-0.1,"O"],
                  [1.25,-0.1,"A (1,0)"],
                  [cos(theta),-0.1,"B"],
                  [cos(theta)*cos(theta)-0.1,cos(theta)*sin(theta)+0.1,"C"],
                  [cos(theta)+0.25,sin(theta)+0.1,"P (cos t, sin t)"],
                  [-0.2,1.1,"(0,1)"],
                  [0.3,0.1,"t"] } ):
    Title := proc( theta )
      if nargs > 1
        then
          if args[2]=`shorttitle`
            then return sprintf( "Unit circle with theta=%07.5f", theta )
            else return ""
          end if
        else
          return sprintf( " When t=%07.5f:\narea( sector(COB) ) =   %07.5f\narea( triangle(POB) ) = %07.5f\narea( sector(POA) ) =   %07.5f",
                            theta,
                            abs(theta)*cos(theta)^2/2,
                            cos(theta)*abs(sin(theta))/2,
                            abs(theta)/2 ):
      end if
    end proc;
  plots[display]( [C,S(theta),A(theta),P(theta),T(theta)],
                  view=[-1.7..1.7,-1.7..1.7],
                  title=Title(args),
                  scaling=constrained ):
end proc:

>

Lesson Overview

Limits of expressions involving the trigonometric functions can be evaluated by direct substitution --- unless division by zero is encountered. When division by zero is encountered, simplifications such as the ones presented in the Conceptual Understanding of Limits lesson can be attempted. In many cases, however, this is not enough to evaluate a limit. The two special limits

limit(sin(theta)/theta,theta = 0) = 1 and limit((1-cos(theta))/theta,theta = 0) = 0 .

were introduced -- graphically -- in the Conceptual Understanding of Limits lesson. In the current lesson each of these limits will be rigorously evaluated. These proofs will be our first real application of the Limit Laws, particularly the Squeeze Theorem.

>

Limit Laws for Trigonometric Functions

The sine and cosine functions are the foundation for all trigonometric functions. These functions are periodic, with period . From the familiar graphs of these two functions

>    Psin := plot( [sin(theta),cos(theta)], theta=-3*Pi..3*Pi, color=[red,blue],
              title="Three periods of the sine and cosine functions",
              legend=["y=sin(theta)", "y=cos(theta)"] ):
Psin;

>

it is easily seen that

The basic limits for the other four trigonometric functions, tangent, cotangent, secant, and cosecant, are obtained from sine and cosine. Based on the Limit Laws, particularly the one for quotients, it is easily seen that many limits involving these functions can be evaluated by substitution.

These formulae can be summarized by saying that limits involving any of these six trigonometric functions can be evaluated by substitution so long as the limit point is in the domain of the function. For the tangent and secant functions, points where the cosine function is zero must be excluded, i.e., odd multiples of ; for the cotangent and cosecant functions, the bad points are the zeros of the sine function, i.e., all (integer) multiples of . The graphs of these six functions can be helpful in remembering the domains of these functions.

>    Ptan := plot( [tan(theta),cot(theta)], theta=-3*Pi..3*Pi, color=[red,blue],
              view=[-3*Pi..3*Pi,-10..10], discont=true,
              title="Six periods of the tangent and cotangent functions",
              legend=["y=tan(theta)", "y=cot(theta)"] ):
Psec := plot( [sec(theta),csc(theta)], theta=-3*Pi..3*Pi, color=[red,blue],
              view=[-3*Pi..3*Pi,-10..10], discont=true,
              title="Three periods of the secant and cosecant functions",
              legend=["y=sec(theta)", "y=csc(theta)"] ):
Psin; Ptan; Psec;   # would like to display side-by-side

The Limit maplet [ Maplet Viewer] [ MapleNet] can be used to evaluate limits involving trigonometric functions. To apply the six trigonometric Limit Laws listed above, select the funtion from the palette that appears when the Select a Function button is pressed or simply enter the name of the trigonometric function ( sin , cos , tan , cot , sec , or csc ) in the box in the Function Rules section of the GUI. Click the Apply button to apply this rule in the current problem.

>

A Rigorous Evaluation of Two Special Limits

In the Conceptual Understanding of Limits lesson graphical arguments are used to justify the statements

limit(sin(theta)/theta,theta = 0) = 1 and limit((1-cos(theta))/theta,theta = 0) = 0 .

While the arguments based on the plots of y = sin(theta)/theta and y = (1-cos(theta))/theta are suggestive, they are not proofs.

To provide rigorous proofs that these limits exist and have the values suggested by the graphs, the Limit Laws, particularly the Squeeze Theorem, will be utilized.

>

Proof of

Consider the following two diagrams.

> figure( Pi/4, `shorttitle` );

> figure( Pi/18, `shorttitle` );

>

The above pictures illustrate that the area of triangle POB is bounded above by the area of sector POA and bounded below by the area of sector COB. When the angle made with the -axis decreases to zero, all three areas also decrease to zero. This is clearly shown in the following animation.

> Theta := [seq(Pi/60*(31-i),i=2..30)]:
Frames := [seq(figure(t),t=Theta)]:
display( Frames, insequence=true );

>

Observe that, for small angles, the three areas appear to be almost equal.

Formulae for the three areas can be found from basic geometry:

area( sector COB ) = =

area( triangle POB ) =

area( sector( POA ) = (1) = .

>

These combine to yield:

<= <=

Assume that and . Then it is permissible to multiply these inequalities by the nonzero quantity and obtain:

<= <=

For all in ( , ) (excluding ), > 0 and so . This means

<= <=

The next plot confirms the correctness -- and usefulness -- of these inequalities.

>    plot( [ 1/cos(theta), sin(theta)/theta, cos(theta) ],
      theta=-1..1, color=[blue,red,green],
      title="Squeeze Theorem for sin(theta)/theta at theta=0",
      legend=["y=1/cos(theta)", "y=sin(theta)/theta", "y=cos(theta)"] );

>

To conclude, because = = 1, the Squeeze Theorem is applicable and tells us that

>

Proof of

This proof begins by multiplying by and then using trigonometric identities to simplify:

Therefore,

= = (1)(0) = 0

>

Lesson Summary

Limits involving the fundamental trigonometric functions sine and cosine are easy to evaluate so long as division by zero is avoided. (In the Continuity lesson we will learn that the sine and cosine functions are continuous for all real numbers.) When division by zero is encountered, simplifications via factorization can be attempted. In many cases, however, this will not be enough to evaluate a limit. In these situations you should remember the two special limits

limit(sin(theta)/theta,theta = 0) = 1 and limit((1-cos(theta))/theta,theta = 0) = 0 .

Even if these expressions do not appear explicitly in the original limit, remember to explore different ways you might be able to manipulate the expression into a form in which one of these limits can be applied.

The methods used in the rigorous evaluation of the two special limits --- geometry and the Squeeze Theorem --- can be used for other problems. Finding appropriate upper and lower bounds for the Squeeze Theorem can be challenge. A graph can sometimes help to suggest reasonable choices for the bounding functions.

>

What's Next?

To conclude this lesson, use the online practice problems to obtain additional experience with the Limit Laws as they are applied for trigonometric problems. When you are confident in yours skills, complete the online homework assignment and the textbook homework assignment.

This is the mid-point in Unit 1. This is a good time to review all of the concepts in the first four lessons: Conceptual Understanding of the Limit, Precise Definition of the Limit, Limit Laws, and Limits of Trigonometric Functions. To test your mastery of these topics, complete the (online) Quiz 1.

The second half of the unit looks at Limits at Infinity, Infinite Limits, and Continuity before concluding with one of the essetial theoretical applications of continuity, the Intermediate Value Theorem. The Limits at Infinity lesson takes a closer look at limits in which the limit point is unbounded, i.e., either or .

>