Limits at Infinity and Horizontal Asymptotes

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Presentation transcript:

Limits at Infinity and Horizontal Asymptotes Lesson 2-6 Limits at Infinity and Horizontal Asymptotes

Objectives Identify and use limits of functions as x approaches either +/- ∞ Identify horizontal asymptotes of functions

Vocabulary Horizontal Asymptote – a line y = L is a horizontal asymptote, if either limx→∞ f(x) = L or limx→-∞ f(x) = L Infinity – ∞ (not a number!! ∞ - ∞ ≠ 0)

Limits at Infinity Horizontal Asymptotes: 10x² + 9 f(x) = ------------- 5x² + 1 16x4 + x² g(x) = ------------- 4x4 + 7 y = 4 y = 2 x² (10 + 9/x²) 10 lim f(x) = lim -------------------- = lim ----- = 2 x² (5 + 1/x²) 5 x4 (16 + 1/x²) 16 lim g(x) = lim -------------------- = lim ----- = 4 x4 (4 + 7/x4) 4 x→∞ x→∞ x→∞ x→-∞ x→-∞ x→-∞ 11x4 + x² h(x) = ------------- 3x2 + 7 x2 (11x2 + 1) 11x² lim h(x) = lim -------------------- = lim ------- = ∞ x2 (3 + 7/x2) 3 x→∞ x→∞ x→∞ lim (x² - 5x) = lim x² - 5 lim x = ∞ not ∞ - ∞ !! x→∞ x→∞ x→∞ Remember infinity is not a number!

Rational Functions When given a ratio of two polynomials, the limit of the function as x approaches infinity will be determined by the ratio of highest powers (HP) of x in numerator and the denominator: HPs equal: then the limit is the ratio of the constants in front of the HP x-terms (and its horizontal asymptote) HP in numerator > HP in denominator: then the limit is DNE (and no horizontal asymptotes exist) HP in numerator < HP in denominator: then the limit is 0 (and the horizontal asymptote is y = 0) 7x³ - 3x² - 2x + 1 7 example: lim -------------------------- = ---- x 4x³ - 13x² + 7 4 5x³ + 7x² - 3x + 4 example: lim -------------------------- = DNE x 3x² - 8x + 5 -6x² - 8x - 7 example: lim ---------------------- = 0 x 2x³ + 7

Horizontal Asymptotes A horizontal asymptote for a function f is a line y = L such that , either , or , or both. A function may have at most 2 horizontal asymptotes. lim f(x) = L x lim f(x) = L x-

Example 1 Evaluate: a. b. c. d. 5x³ + 7x + 1 lim ----------------------- = x 3x³ + 2x² + 3 2x + 5 lim --------------- = x  x² + 4 cos x lim --------------- = x x x³ + 6x + 1 lim --------------------- = x 2x² - 5x

Example 2 Find the horizontal asymptote(s) for x² - 2x + 1 y = ------------------- 3x³ + 4x + 7 3x7 – 4x5 + 3x - 1 y = -------------------------- 2x7 + 4 x y = ---------------- x² - 1

Checking for Understanding

Summary & Homework Summary: Limits at infinity involved the highest powers in the function Horizontal asymptotes (y = L) are the limits that exist (as x approaches infinity) Homework: pg 146 - 149: 2, 3, 7, 11, 13, 18, 27, 29, 33, 38, 39