Limits Of Trigonometric Functions.pdf

72 2 Limits and Continuity of Functions 2 Limits o f rigonometric unctions Theorem 2.18) y x FIGURE 2 17 Corollary 2.19) Whenever we discuss limits of trigonometric expressions involving sin t cos x, tan 0 etc., we shall assume that each variable represents a real number or the radian measure o f an angle. The following result is important for future developments. [ Jim sin t = 0 r-o Proof Let us first prove that lim 1 _ 0 sin t = 0. Since we are only interested in positive values of t near zero, there is no loss of generality in assuming that 0 < t < n/2. Let U be the circl e of radius 1 with center at the origin of a rectangular coordinate system, and let A be the point 1, 0 . If, as illustrated in Figure 2.17, P x, y is the point on U that corresponds to t then by 1.26 ) the radian measure of angle OP is t. Referring to the figure we see that O y t or, since y = sin 0 <sin t < t. Since lim 1 0 t = 0, it follo ws from the Sandwich Theore m 2.14 that lim 1 _ 0 sin t = 0. To complete the proof it is sufficient to show that lim 1 _ 0 _ sin t = 0. f - n/2 < t < 0, then 0 < t < n/2 and hence, from the first part of the proof, 0 < sin - t < - t. Multiplying the last inequality by 1 and using the fact that sin - t = - sin t gives us t <sin t < 0. Since lim 1 _ 0 _ t = 0, it follows from the Sandwich Theorem that lim 1 _ 0 _ sin t = 0. D Jim cost= 1 r-o Proof Since sin 2 t cos 2 t = 1, it follows that cost = ± l sin 2 t. f n/2 < t < n/2, then cost is positive, and hence cost= J 1 - sin 2 t. Consequently, Jim cost= limJl - sin 2 t = J im(l - sin 2 t r o t o r o = j1--=o = 1. D

Limits Of Trigonometric Functions.pdf

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