Continuity of Trig and Inverse Functions Objective: To use limits to define continuity in trig/inverse functions Theorem 1.6.1 • If c is any number in the natural domain of the stated trigonometric function, then lim sin x sin c lim cos x cos c lim tan x tan c lim cot x cot c x c x c lim sec x sec c lim csc x csc c x c x c x c x c Example 1 • Find the limit x2 1 lim cos x 1 x 1 Example 1 • Find the limit x2 1 lim cos x 1 x 1 • The limit of the cosine is the cosine of the limit. x2 1 x2 1 cos lim cos lim ( x 1) cos 2 lim cos x 1 x 1 x 1 x 1 x 1 Theorem 1.6.2 • If f is a one-to-one function that is continuous at each point of its domain, then f -1 is continuous at each point of its domain, that is f -1 is continuous at each point of the range of f. Example 3 • Where is the function continuous? tan 1 x ln x f ( x) x2 4 Example 3 • Where is the function continuous? tan 1 x ln x f ( x) x2 4 • A fraction is continuous where the numerator and denominator are continuous and the denominator is not zero. Example 3 • Where is the function continuous? tan 1 x ln x f ( x) x2 4 • A fraction is continuous where the numerator and denominator are continuous and the denominator is not zero. • The numerator is continuous for x > 0 (why?) and the denominator is continuous everywhere x 2 Example 3 • Where is the function continuous? tan 1 x ln x f ( x) x2 4 • A fraction is continuous where the numerator and denominator are continuous and the denominator is not zero. • The numerator is continuous for x > 0 (why?) and the denominator is continuous everywhere x 2 • The function f is continuous for x > 0, not 2. The Squeezing Theorem • Let f, g, and h be functions satisfying g ( x ) f ( x ) h( x ) for all x in some open interval containing the number c, with the possible exception that the inequalities need not hold at c. If g and h have the same limit as x approaches c, say lim g ( x) lim h( x) L x c x c then f also has this limit f ( x) L as x approaches c, that is lim x c Theorem 1.6.5 sin x ( a ) lim 1 x 0 x 1 cos x (b) lim 0 x 0 x Example 4 • Find: tan x lim x 0 x Example 4 • Find: tan x lim x 0 x sin x 1 sin x 1 lim lim lim 1 1 1 x 0 x 0 x 0 x cos x x cos x Example 4 • Find: sin 2 x lim x 0 x Example 4 • Find: sin 2 x lim x 0 x 2 sin 2 x sin 2 x lim 2 lim 2 1 2 x 0 x 0 2x 2x Example 4 • Find: sin 3 x lim x 0 sin 5 x Example 4 • Find: sin 3 x lim x 0 sin 5 x sin 3 x lim x x 0 sin 5 x x Example 4 • Find: sin 3 x lim x 0 sin 5 x sin 3x sin 3x 3 3x lim x lim x 0 sin 5 x x 0 sin 5 x 5 x 5x Example 4 • Find: sin 3 x lim x 0 sin 5 x sin 3x sin 3x 3 3 x 3 x lim lim lim x 0 sin 5 x x 0 sin 5 x 5 x0 5 x 5x 3 1 3 5 1 5 sin 3x 3x sin 5 x 5x Homework • Section 1.6 • Pages125-126 • 1-35 odd