1. Limits Calculus 1.1 and 1.2

2. Derivatives Problem: Find the area of this picture. 9/18/2015 – LO: Limits - Determine if they exist. #102 p54 5, 8, 12, 15-24, 26, 28

3. Intro to Calculus Derivatives Problem: Find the area of this picture.

4. Intro to Calculus Derivatives Problem: Find the area of this picture.

5. The Tangent Line Problem

6. x ∆x = distance from x x + ∆x 1 ∆x = 4 5 y f(x)f(x) y + ∆y f(x+∆x) f(1)f(1) f(5)f(5) ∆y =2 =10 ∆y = 8

7. The Tangent Line Problem f( 1 + 4 )

8. The Tangent Line Problem xx xx

9. Develop a habit to try 3 approaches to problem solving. 1. Numerical Approach Construct a table of values. 2. Graphical Approach Draw a graph. 3. Analytical Approach Use algebra or calculus.

10. 3 Limits that FAIL 1. Behavior that is different from Right and Left As x creeps to the limit, the function goes towards different values from the right and the left.

11. 3 Limits that FAIL 2. Unbounded Behavior Function goes to positive or negative infinite.

12. 3 Limits that FAIL 3. Oscillating Behavior Trig functions with x in the denominator.

13. Limits that FAIL Common types of behavior where the limit does not exist. f(x) approaches different values from left and right. f(x) increase or decrease without bound as x approaches c. f(x) oscillates between 2 different values as x approaches c.

14. 0.2040.20040.20.19990.19960.196 = 0.2 1.Put function into Graph mode (MENU 5). 2.Verify appropriate window (V-Window) 3.Draw graph - Analyze graph at limit 4.Use TRACE to complete table

16. 1.Put function into Graph mode (MENU 5). 2.Verify RAD mode (Shift - SETUP) 3.V-Window to TRIG 4.Draw graph - Analyze graph at limit 5.Use TRACE to complete table 0.998330.999980.99999 0.999980.99833 = 1

18. x f(x)f(x) 10.9990.990.91.0011.011.1 IND F 0.66730.67330.73400.66600.66000.6015 = 0.667= 2/3

19. = 1= 4 = 2 = 4

20. Limit does not exist. The function approaches 1 from the right and -1 from the left as x approaches 2. Limit does not exist. The function decreases from the left without bound and increases from the right without bound as x approaches 5.

22. Yes.f(1) = 2 (b) No, the limit does not exist. As the x approaches 1, the function approaches 1 from the right but approaches 3.5 from the left. (c) No, the value does not exist. The function is undefined when x = 4. (d) Yes. f(4) = 2.

24. At x = -3, the function does not exist. At x = 2, the function DOES exist.