Section 1-4 - LimitTrig

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Chapter 1: Limits

Section 1.4: Limits for Trig Functions

Essentials

Two Special Trig Limits

Special Trig Limit 1	$lim_{θ \to 0} \frac{\sin (θ)}{θ} = 1$
Special Trig Limit 2	$lim_{θ \to 0} \frac{1 - \cos (θ)}{θ} = 0$
Table 1.4.1 Special trig limits from Section 1.1

Graphical justification for these limits is provided by Figures 1.4.1 and 1.4.2.

Figure 1.4.1 Graph of $\frac{\sin (θ)}{θ}$ on $[- 4 π, 4 π]$

Figure 1.4.2 Graph of $\frac{1 - \cos (θ)}{θ}$ on $[- 4 π, 4 π]$

Each function is undefined at $θ = 0$ because of division by zero. On each graph, the black dot represents the "hole" where $θ = 0$ , the first at $(0, 1)$ ; and the second, at $(0, 0)$ . However, the graphs also show that near $θ = 0$ each function is defined, and from these nearby values the limits can be estimated.

Unfortunately, no simple application of the "factor and cancel" and "rationalize the numerator" strategies will establish the first of these two limits. The next subsection, however, provides the appropriate valid proofs.

Proofs for Limits in Table 1.4.1

Special Trig Limit 1

To establish $lim_{θ \to 0} \frac{\sin (θ)}{θ} = 1$ , apply the Squeeze theorem to the inequality

$\cos (θ) \leq \frac{\sin (θ)}{θ}$ $\leq \frac{1}{\cos (θ)}$

Since the left and right sides approach 1, the middle fraction must also approach 1. But where did this inequality come from? A comparison of the three areas shown in Figure 1.4.3, and delineated in Table 1.4.2, is the basis of this inequality. (Interpret the letter "t" in Figures 1.4.3 and 1.4.5 as the angle $θ$ .)

figure := proc( theta )
  local A, C, P, S, T, Title, t;
  C := plots:-implicitplot( x^2+y^2=1, x=-1..1, y=-1..1, tickmarks=[2,2] ):
  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 -> plots:-pointplot( [ [1,0],
                           [cos(theta),0],
                           [cos(theta)*cos(theta),cos(theta)*sin(theta)],
                           [cos(theta),sin(theta)] ],
                           symbol=circle, symbolsize=12 ):
  T := theta -> plots:-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 t = %07.5f", theta )
            else return ""
          end if
        else
          return sprintf( " When t=%07.5f:
                                area( sector(COB) ) = %07.5f
                                area( triangle(POB) ) = %07.5f
                                area( 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:

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

area( sector COB )	$= \frac{\|θ\|}{2 π}$ $π \cos^{2} (θ)$ = $\frac{\|θ\|}{2}$ $\cos^{2} (θ)$
area( triangle POB )	$= \frac{1}{2}$ $\cos (θ) \|\sin (θ)\|$
area( sector POA )	$= \frac{\|θ\|}{2 π}$ $π ($ $1)^{2}$ = $\frac{\|θ\|}{2}$

Figure 1.4.3 Unit circle, with ΔPOB, and sectors POA and COB

Table 1.4.2 Expressions for areas of ΔPOB, and sectors POA and COB

The relationship of the three areas detailed in Table 1.4.2 and inferred from Figure 1.4.3, is given by the inequality

$\frac{|θ|}{2}$ $\cos^{2} (θ) \leq \frac{1}{2}$ $\cos (θ) \sin (θ) \leq \frac{|θ|}{2}$

For $θ$ small but nonzero so that $\cos (θ) > 0$ , multiply through by the positive quantity $\frac{2}{|θ| \cos (θ)}$ to obtain

$\cos (θ) \leq \frac{|\sin (θ)|}{|θ|}$ $\leq$ $\frac{1}{\cos (θ)}$

For nonzero $θ$ in $(- π / 2, π / 2)$ , $\frac{\sin (θ)}{θ} > 0$ and $\frac{|\sin (θ)|}{|θ|} = \frac{\sin (θ)}{θ}$ . Hence, the key inequality

$\cos (θ) \leq \frac{\sin (θ)}{θ}$ $\leq \frac{1}{\cos (θ)}$

is obtained, and from which follows $lim_{θ \to 0} \frac{\sin (θ)}{θ} = 1$ by an application of the Squeeze theorem.

Indeed, Figure 1.4.4 compares graphs of the three terms in the key inequality, and the animation in Figure 1.4.5 shows the dynamic relationship between the areas listed in Table 1.4.2.

>	plot([cos(x), sin(x)/x, 1/cos(x)], x=-1..1, color=[blue,red,green], legend =[typeset(cos(x)), typeset(sin(x)/x), typeset(1/cos(x))] );

figure := proc( theta )
  local A, C, P, S, T, Title, t;
  C := plots:-implicitplot( x^2+y^2=1, x=-1..1, y=-1..1, tickmarks=[2,2] ):
  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 -> plots:-pointplot( [ [1,0],
                           [cos(theta),0],
                           [cos(theta)*cos(theta),cos(theta)*sin(theta)],
                           [cos(theta),sin(theta)] ],
                           symbol=circle, symbolsize=12 ):
  T := theta -> plots:-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)+0.7,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:
                                area( sector(COB) ) = %07.5f
                                area( triangle(POB) ) = %07.5f
                                area( 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:

>	Theta := [seq(Pi/60*(31-i),i=2..30)]: Frames := [seq(figure(t),t=Theta)]: plots:-display( Frames, insequence=true,size=[600,400] );

Figure 1.4.4 Graphs of $\cos (x), \frac{\sin (x)}{x}, \frac{1}{\cos (x)}$

Figure 1.4.5 Animation showing relationship between areas of ΔPOB, and sectors COB and POA

Special Trig Limit 2

To establish $lim_{θ \to 0} \frac{1 - \cos (θ)}{θ} = 0$ , multiply $\frac{1 - \cos (θ)}{θ}$ by $1 = \frac{1 + \cos (θ)}{1 + \cos (θ)}$ and then use trigonometric identities to simplify. The steps are

$\frac{1 - \cos (θ)}{θ}$	= $\frac{1 - \cos (θ)}{θ}$ $\frac{1 + \cos (θ)}{1 + \cos (θ)}$
	= $\frac{1 - \cos^{2} θ}{θ (1 + \cos (θ))}$
	= $\frac{\sin^{2} θ}{θ (1 + \cos (θ))}$
	= $\frac{\sin (θ)}{θ}$ $\frac{\sin (θ)}{1 + \cos (θ)}$

Consequently,

$lim_{θ \to 0} \frac{1 - \cos (θ)}{θ}$ = $(lim_{θ \to 0} \frac{\sin (θ)}{θ}) (lim_{θ \to 0} \frac{\sin (θ)}{1 + \cos (θ)})$ = (1)(0) = 0

General Limit Laws for Trigonometric Functions

From Figures 1.4.6 - 8, Table 1.4.3 can be inferred.

>	plot([sin(x),cos(x)], x=-3Pi..3Pi, color=[red,blue],legend=[typeset(sin(x)), typeset(cos(x))]);

>	plot([tan(x),cot(x)], x=-3Pi..3Pi, color=[red,blue], view=[-3Pi..3Pi,-10..10], discont=true, legend=[typeset(tan(x)), typeset(cot(x))]);

>	plot([sec(x),csc(x)], x=-3Pi..3Pi, color=[red,blue], view=[-3Pi..3Pi,-10..10], discont=true, legend=[typeset(sec(x)), typeset(csc(x))]);

Figure 1.4.6 Three periods of the sine and cosine functions

Figure 1.4.7 Six periods of the tangent and cotangent functions

Figure 1.4.8 Three periods of the secant and cosecant functions

Table 1.4.3 lists where the elementary trig functions are undefined because they have vertical asymptotes at these locations. For all other values than the restricted values, the limit of one of these functions can be found by simply evaluating the function.

Limit	Restrictions on c
$lim_{θ \to c} \sin (θ) = \sin (c)$	none
$lim_{θ \to c} \cos (θ) = \cos (c)$	none
$lim_{θ \to c} \tan (θ) = \tan (c)$	$c \neq \frac{(2 n + 1) π}{2}$
$lim_{θ \to c} \cot (θ) = \cot (c)$	$c \neq n π$
$lim_{θ \to c} \sec (θ) = \sec (c)$	$c \neq \frac{(2 n + 1) π}{2}$
$lim_{θ \to c} \csc (θ) = \csc (c)$	$c \neq n π$
Table 1.4.3 Limits of the elementary trig functions at points $c$ in their domains; $n$ is any integer

Table 1.4.3 does not address the limits of the elementary trig functions at points not in their domains. That must await the material in Section 1.6.

Précis

The sine and cosine functions are defined for all real numbers. The limit of either function at any real number is the value of the function. However, for the other four elementary trig functions, there are an infinite number of points in the domains where the functions are not defined. At these points, the functions have vertical asymptotes. For all other real numbers, the function values are the limits.

The two special limits in Table 1.4.1 take some care to establish. Special Trig Limit 1 does not yield to the "factor and cancel" strategy, but rather, requires an application of the Squeeze theorem based on a shrewd geometric analysis. Special Trig Limit 2 actually yields to the "rationalize the numerator" strategy, provided Special Trig Limit 1 has already been established.

Examples

Example 1.4.1	Use the contents of Table 1.4.1 to evaluate $lim_{x \to 0} \frac{\sin (3 x)}{4 x}$ .
Example 1.4.2	Justify the expression for the area of sector COB in Table 1.4.2.
Example 1.4.3	Justify the expression for the area of triangle POB in Table 1.4.2.
Example 1.4.4	Justify the expression for the area of sector POA in Table 1.4.2.

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