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Teaching Tip Alternate Example Finding the Limit of a Trigonometric Function x x+ x Find lim 2 csc 2 csc . x→0 2 xcsc x Solution This time, we rewrite in terms of sines: x x+ x lim 2 csc 2 csc x→0 2 xcsc x ⎛ 2x 1 ⎞ ⎜ 2 ⎟ = lim sin x ⎜ + sin x ⎟ x→0 ⎜ x x ⎟ ⎝ 2 2 sin x sin x ⎠ ⎛ x x x ⎞ = lim ⎜ 2 sin 2 2 sin + x→0 2 ⎟ ⎝ xsin x xsin x⎠ ⎛ sin x ⎞ = lim + ⎝ ⎜ 2 ⎠ ⎟ = 2 + 1 = 3 x→0 x Chapter objective 3: Determine where the trigonometric functions are continuous is a good review. This knowledge is a calculus prerequisite, so if the students are well prepared, you may be able to address this objective briefly to save time. AP® CaLC skill builder for example 4 Finding the Limit of a Trigonometric Function Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 122 Chapter 1 • Limits and Continuity 2π π π π π 2 2 y 1 1 y 1 1 y cos x y sin x 3π 2 2π π π π π 3π 2π x 2 2 2 Figure 45 x Sullivan Solution First we rewrite the expression cos θ − 1 as the product of two terms whose θ limits are known. For θ = 0, ( )( ) cos θ − 1 cos θ − 1 cos θ + 1 = θ θ cos θ + 1 = cos2 θ − 1 θ(cos θ + 1) = − sin 2 ( ) θ sin θ (− sin θ) ↑ θ(cos θ + 1) = θ cos θ + 1 sin 2 θ+ cos 2 θ = 1 Now we find the limit. [( )( )] [ ][ ] cos θ − 1 sin θ (− sin θ) sin θ − sin θ lim = lim = lim lim θ→0 θ θ→0 θ cos θ + 1 θ→0 θ θ→0 cos θ + 1 lim(− sin θ) θ→0 = 1 · lim (cos θ + 1) = 0 2 = 0 ■ θ→0 NOW WORK Problem 31 and AP® Practice Problem 8. 3 Determine Where the Trigonometric Functions Are Continuous The graphs of f (x) = sin x and g(x) = cos x in Figure 45 suggest that f and g are continuous on their domains, the set of all real numbers. EXAMPLE 4 Showing f(x) = sin x Is Continuous at 0 • f (0) = sin 0 = 0, so f is defined at 0. • lim f (x) = lim sin x = 0, so the limit at 0 exists. x→0 x→0 • lim sin x = sin 0 = 0. x→0 Since all three conditions of continuity are satisfied, f (x) = sin x is continuous at 0. ■ In a similar way, we can show that g(x) = cos x is continuous at 0. That is, lim cos x = cos 0 = 1. x→0 You are asked to prove the following theorem in Problem 67. THEOREM • The sine function y = sin x is continuous on its domain, all real numbers. • The cosine function y = cos x is continuous on its domain, all real numbers. Based on this theorem the following two limits can be added to the list of basic limits. lim sin x = sin c for all real numbers c x→c lim cos x = cos c for all real numbers c x→c Following are some function values for two continuous functions, f and g. x fx ( ) gx ( ) 1 2 −1 2 1 0 sin( gx ( )) + fx ( ) Find lim . x→2 gf (( x)) Solution Here we will use the continuity of f, g, and the sine function to find the limit. sin( gx ( )) + fx ( ) lim x→2 gf (( x)) ∑ Mathematical Practices Tip MPAC 1: Reasoning with Definitions and Theorems Use the continuity of a function to find a limit. Ask students to use the definition of continuity to explain why lim cosx =−1. Have students x→π explain by using a graph how the continuity of a function f at a point x = c shows that lim f( x) = f( c). x→c sin( g(2)) + f(2) = lim x→2 gf ((2)) + + = lim sin(0) 1 0 1 = − =− 1 x→2 g(1) 1 122 Chapter 1 • Limits and Continuity TE_Sullivan_Chapter01_PART II.indd 5 11/01/17 9:55 am
Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Section 1.4 • Limits and Continuity of Trigonometric, Exponential, and Logarithmic Functions 123 Using the facts that the sine and cosine functions are continuous for all real numbers, we can use basic trigonometric identities to determine where the remaining four trigonometric functions are continuous: • y = tan x: Since tan x = sin x , from the Continuity of a Quotient property, cos x y = tan x is continuous at all real numbers except those for which cos x = 0. That is, y = tan x is continuous on its domain, all real numbers except odd multiples of π 2 . • y = sec x: Since sec x = 1 , from the Continuity of a Quotient property, cos x y = sec x is continuous at all real numbers except those for which cos x = 0. That is, y = sec x is continuous on its domain, all real numbers except odd multiples of π 2 . • y = cot x: Since cot x = cos x , from the Continuity of a Quotient property, sin x y = cot x is continuous at all real numbers except those for which sin x = 0. That is, y = cot x is continuous on its domain, all real numbers except integer multiples of π. • y = csc x: Since csc x = 1 , from the Continuity of a Quotient property, sin x y = csc x is continuous at all real numbers except those for which sin x = 0. That is, y = csc x is continuous on its domain, all real numbers except integer multiples of π. Teaching Tip The abbreviation of the function as “cos” is derived from the word “cosine,” which is also an abbreviation—of the longer name “complement of sine.” In a right triangle, the cosine of an angle is the same as the sine of the complementary angle. The same explanation holds for cotangent and cosecant. The cotangent of an angle is the same as the tangent of the complementary angle, and the cosecant of an angle is the same as the secant of the complementary angle. NEED TO REVIEW? Inverse trigonometric functions are discussed in Section P.7, pp. 61--66. Recall that a one-to-one function that is continuous on its domain has an inverse function that is continuous on its domain. Since each of the six trigonometric functions is continuous on its domain, then each is continuous on the restricted domain used to define its inverse trigonometric function. This means the inverse trigonometric functions are continuous on their domains. These results are summarized in Table 10. TABLE 10 Continuity of the Trigonometric Functions and Their Inverses Function Domain Properties Sine all real numbers continuous on the interval (−∞, ∞) Cosine all real numbers continuous on the interval (−∞, ∞) { Tangent x|x = odd multiples of π } continuous at all real numbers except odd multiples of π 2 2 Cosecant {x|x = multiples of π} continuous at all real numbers except multiples of π { Secant x|x = odd multiples of π } continuous at all real numbers except odd multiples of π 2 2 Cotangent {x|x = multiples of π} continuous at all real numbers except multiples of π Inverse sine −1 ≤ x ≤ 1 continuous on the closed interval [−1, 1] Inverse cosine −1 ≤ x ≤ 1 continuous on the closed interval [−1, 1] Inverse tangent all real numbers continuous on the interval (−∞, ∞) Inverse cosecant |x| ≥1 continuous on the set (−∞, −1] ∪ [1, ∞) Inverse secant |x| ≥1 continuous on the set (−∞, −1] ∪ [1, ∞) Inverse cotangent all real numbers continuous on the interval (−∞, ∞) Section 1.4 • Limits and Continuity of Trigonometric, Exponential, and Logarithmic Functions 123 TE_Sullivan_Chapter01_PART II.indd 6 11/01/17 9:55 am
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