Chapter 1: Limits
Section 1.4: Limits for Trig Functions
Essentials
Two Special Trig Limits
Special Trig Limit 1
limθ→0sinθθ=1
Special Trig Limit 2
limθ→01−cosθθ=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 sinθθ on −4 π,4 π
Figure 1.4.2 Graph of 1−cosθθ 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
To establish limθ→0sinθθ=1, apply the Squeeze theorem to the inequality
cosθ≤sinθθ≤1cosθ
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 )
=θ2π πcos2θ = θ2cos2θ
area( triangle POB )
=12cosθsinθ
area( sector POA )
=θ2π π (1)2 = θ2
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
θ2cos2θ≤12cosθsinθ≤θ2
For θ small but nonzero so that cosθ>0, multiply through by the positive quantity 2θcosθ to obtain
cosθ≤sinθθ≤1cosθ
For nonzero θ in −π/2,π/2, sinθθ>0 and sinθθ=sinθθ. Hence, the key inequality
is obtained, and from which follows limθ→0sinθθ=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 cosx,sinxx,1cosx
Figure 1.4.5 Animation showing relationship between areas of ΔPOB, and sectors COB and POA
To establish limθ→01−cosθθ=0, multiply 1−cosθθ by 1=1+cosθ1+cosθ and then use trigonometric identities to simplify. The steps are
1−cosθθ
= 1−cosθθ 1+cosθ1+cosθ
= 1−cos2θθ1+cosθ
= sin2θθ1+cosθ
= sinθθ sinθ1+cosθ
Consequently,
limθ→01−cosθθ = limθ→0sinθθlimθ→0sinθ1+cosθ = (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=-3*Pi..3*Pi, color=[red,blue],legend=[typeset(sin(x)), typeset(cos(x))]);
plot([tan(x),cot(x)], x=-3*Pi..3*Pi, color=[red,blue], view=[-3*Pi..3*Pi,-10..10], discont=true, legend=[typeset(tan(x)), typeset(cot(x))]);
plot([sec(x),csc(x)], x=-3*Pi..3*Pi, color=[red,blue], view=[-3*Pi..3*Pi,-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
limθ→csinθ=sinc
none
limθ→ccosθ=cosc
limθ→ctanθ=tanc
c≠2n+1π2
limθ→ccotθ=cotc
c≠nπ
limθ→csecθ=secc
limθ→ccscθ=cscc
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 limx→0sin3 x4 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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