1. Lesson 2-2 The Limit of a Function

2. 5-Minute Check on Algebra Transparency 1-1 Click the mouse button or press the Space Bar to display the answers. 1.6x + 45 = 18 – 3x 2.x 2 – 45 = 4 3.(3x + 4) + (4x – 7) = 11 4.(4x – 10) + (6x +30) = 180 5.Find the slope of the line k. 6. Find the slope of a perpendicular line to k Standardized Test Practice: y x k A (0,1) (-6,-2) B (6,4) C ACBD 1/22-1/2 -2

3. 5-Minute Check on Algebra Transparency 1-1 Click the mouse button or press the Space Bar to display the answers. 1.6x + 45 = 18 – 3x 2.x 2 – 45 = 4 3.(3x + 4) + (4x – 7) = 11 4.(4x – 10) + (6x +30) = 180 5.Find the slope of the line k. 6. Find the slope of a perpendicular line to k Standardized Test Practice: 9x +45 = 18 9x = -27 x = -3 x² = 49 x = √49 x = +/- 7 7x - 3 = 11 7x = 14 x = 2 10x + 20 = 180 10x = 160 x = 16 ∆y y 2 – y 1 4 – 1 3 1 m = ----- = ----------- = -------- = ------ = ---- ∆x x 2 – x 1 6 – 0 6 2 ACBD 1/22-1/2 -2 y x k A (0,1) (-6,-2) B (6,4) C ∆x ∆y

4. Objectives Determine and Understand one-sided limits Determine and Understand two-sided limits

5. Vocabulary Limit (two sided) – as x approaches a value a, f(x) approaches a value L Left-hand (side) Limit – as x approaches a value a from the negative side, f(x) approaches a value L Right-hand (side) Limit – as x approaches a value a from the positive side, f(x) approaches a value L DNE – does not exist (either a limit increase/decreases without bound or the two one-sided limits are not equal) Infinity – increases (+∞) without bound or decreases (-∞) without bound [NOT a number!!] Vertical Asymptote – at x = a because a limit as x approaches a either increases or decreases without bound

6. Homework Problem # 1 tVm secant 01000-50 5694-44.4Estimates using 10444-38.8 10&205&250&30 15250-33.3 -33.3333 20111-27.8 2528-22.2 300-16.6667

7. When we look at the limit below, we examine the f(x) values as x gets very close to a: read: the limit of f(x), as x approaches a, equals L One-Sided Limits: Left-hand limit (as x approaches a from the left side – smaller) RIght-hand limit (as x approaches a from the right side – larger) The two-sided limit (first one shown) = L if and only if both one-sided limits = L if and only if and lim f(x) = L x  a lim f(x) = L x  a - lim f(x) = L x  a + lim f(x) = L x  a lim f(x) = L x  a - lim f(x) = L x  a + Limits

8. Vertical Asymptotes The line x = a is called a vertical asymptote of y = f(x) if at least one of the following is true: lim f(x) = ∞ x  a lim f(x) = ∞ x  a - lim f(x) = ∞ x  a + lim f(x) = -∞ x  a lim f(x) = -∞ x  a - lim f(x) = -∞ x  a +

9. y x Usually a reasonable guess would be: lim f(x) = f(a) x  a (this will be true for continuous functions) ex: lim f(x) = 2 x  2 but, lim f(x) = 7 x  5 (not f(5) = 1) and lim f(x) = DNE x  16 (DNE = does not exist) 251015 One Sided Limits Limit from right: lim f(x) = 5 x  10 + Limit from left: lim f(x) = 3 x  10 - Since the two one- sided limits are not equal, then lim f(x) = DNE x  10 When we look at the limit below, we examine the f(x) values as x gets very close to a: lim f(x) x  a Limits Using Graphs

10. Example 1 Answer each using the graph to the right (from Study Guide that accompanies Single Variable Calculus by Stewart) a. b. c. d. Lim f(x) = x→ -5 Lim f(x) = x→ 2 Lim f(x) = x→ 0 Lim f(x) = x→ 4 4 3 DNE 0

11. Example 2 Use tables to estimate sin x Lim ------------ = x x→ 0

12. Example 3 Use algebra to find: a. b. c. x³ - 1 Lim ------------ = x - 1 x→ 1 x - 1 Lim ------------ =  x - 1 x→ 1 x 1 Lim --------- - -------- = x – 1 x – 1 x→ 1 Lim (x² + x + 1) = 3 x→ 1 Lim (  x + 1) = 2 x→ 1 Lim 1 = 1 x→ 1

13. Summary & Homework Summary: –Try to find the limit via direct substitution –Use algebra to simplify into useable form Homework: pg 102-104: 5, 6, 7, 9;