1. 2.1 FINITE LIMITS One Sided Limits, Double Sided Limits and Essential Discontinuities www.mathgotserved.com1 Mathgotserved.com

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3. Geometric View of Limits Look at a polygon inscribed in a circle As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.

4. If we refer to the polygon as an n-gon, where n is the number of sides we can make some mathematical statements: As n gets larger, the n-gon gets closer to being a circle As n approaches infinity, the n-gon approaches the circle The limit of the n-gon, as n goes to infinity is the circle The symbolic statement is: Note: The n-gon approaches a circle in appearance even though it is not a circle. It might as well be a circle.

5. Definition (verbal) The limit of a function is the value the function approaches as the independent variable approaches a value, negative or positive infinity. Or think of it as the y value the function approaches as you approach a specific x value. The Limit of a Function Analytical Representation “Read as the limit of f(x) as x approaches a is L” Can also be expressed as f(x) →L as x→a www.mathgotserved.com5

6. Properties of Limits 1. Sum Rule: 2. Difference Rule: 3.Product Rule: 4. Quotient Rule: 5. Constant Rule: 6. Power Rule

7. WHY LEARN LIMITS? Limits are used in derivatives when mathematicians are calculating the velocity of an object in flight. Real World Application : Calculating Speed(Mathematicians: 42-145k/yr) www.mathgotserved.com7

8. One-Sided Limits (Verbal) Numbers x near c fall into two natural categories: those that lie to the left of c and those that lie to the right of c. We write [The left-hand limit of f(x) as x tends to c is L.] to indicate that as x approaches c from the left, f(x) approaches L. [The right-hand limit of f(x) as x tends to c is L.] to indicate that as x approaches c from the right, f(x) approaches L www.mathgotserved.com8 We write

9. How close can Raul Get to the garage? www.mathgotserved.com9

10. 10www.mathgotserved.com Definition of a Limit Left hand Limit: We say the limit as x approaches a from the left isL Right hand Limit: Double sided Limits L We say the limit as x approaches a is L

11. Procedures for Finding Finite Limits Algebraically

12. 12www.mathgotserved.com Finding Finite Limits Algebraically Determine the limit of the following

13. www.mathgotserved.com13 Problem 4: Determine the limit Solution:

14. www.mathgotserved.com14 Problem 5: Determine the limit Solution:

15. Procedure for Finding Graphical Limits Point: Find the y coordinate of the point that you are approaching from that direction. Horizontal Asymptote: Find the y coordinate of the equation of the line Vertical Asymptote: One sided +/- infinity, Double sided does note exist.

16. Problem 1: From both sides you are approaching the point (1,0). The limit is the y coordinate so Solution

17. www.mathgotserved.com17 Finding Limits Graphically Problem 2: Use the graph to find the following

18. www.mathgotserved.com18 f(2) = f(4) = Problem 3 dne 4 4 4 und 1 2 2

19. www.mathgotserved.com19 Finding Finite Limits Numerically Problem 1: Find the following numerically using the table of values xf (x) 45.001 35.0001 234 15.0001 05.001 5.01

20. www.mathgotserved.com20 Problem 2: Find the following limits Numerically xf (x) 2.02-9.99 2.01-9.999 2Error 1.992.9999 1.982.999 1.972.99

21. www.mathgotserved.com21 Problem 3: Find the limit

22. www.mathgotserved.com22 Problem 4: Find the limit