Understanding Limits of Functions & Vertical Asymptotes

1. Lesson 2-2 The Limit of a Function

2. Transparency 1-1 y x k (6,4) C (0,1) (-6,-2) A B 5-Minute Check on Algebra • 6x + 45 = 18 – 3x • x2 – 45 = 4 • (3x + 4) + (4x – 7) = 11 • (4x –10) + (6x +30) = 180 • Find the slope of the line k. • Find the slope of a perpendicular line to k Standardized Test Practice: A B C D -2 1/2 2 -1/2 Click the mouse button or press the Space Bar to display the answers.

3. Transparency 1-1 y x k (6,4) C (0,1) (-6,-2) A B 5-Minute Check on Algebra • 6x + 45 = 18 – 3x • x2 – 45 = 4 • (3x + 4) + (4x – 7) = 11 • (4x –10) + (6x +30) = 180 • Find the slope of the line k. • Find the slope of a perpendicular line to k 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 y2 – y1 4 – 1 3 1 m = ----- = ----------- = -------- = ------ = ---- ∆x x2 – x1 6 – 0 6 2 ∆y ∆x Standardized Test Practice: A B C D -2 1/2 2 -1/2 Click the mouse button or press the Space Bar to display the answers.

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

7. Limits 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+

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 Limits Using Graphs 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 Usually a reasonableguess would be: lim f(x) = f(a) xa (this will be true forcontinuous 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) 2 5 10 15 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

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

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

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