1. 1 § 2-1 Limits The student will learn about: limits, infinite limits, and uses for limits. limits, finding limits, one-sided limits, properties of limits,

2. 2 Procedure During the next few days we will be studying limits. Limits form the foundation of calculus. We will begin by studying limits from a graphical viewpoint. We will end with the importance of limits. We will then look at the properties of limits. We will move on to the algebraic techniques needed to determine limits. 2

3. 3 Functions and Graphs A Brief Review The graph of a function is the graph of the set of all ordered pairs that satisfy the function. As an example the following graph and table represent the same function f (x) = 2x – 1. xf(x) -2-5 -3 0 11 2? 35 We will use this point on the next slide.

4. 4 Limits (THIS IS IMPORTANT) Analyzing a limit. We can also examine what occurs at a particular point. Using the above function, f (x) = 2x – 1, let’s examine what happens when x = 2 through the following table: x1.51.91.991.99922.0012.012.12.5 f (x)24 2 3 2.8 3.2 2.98 3.02 2.998 3.002? Note; As x approaches 2, f (x) approaches 3. This is a dynamic situation

5. 5 Limits IMPORTANT! This table shows what f (x) is doing as x approaches 2. Or we have the limit of the function as x approaches We write this procedure with the following notation. x1.51.91.991.99922.0012.012.12.5 f (x)22.82.982.998?3.0023.023.24 Def: We write if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c). or as x → c, then f (x) → L 2 3 H

6. 6 One-Sided Limit This idea introduces the idea of one-sided limits. We write and call K the limit from the left (or left- hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. 5

7. 7 One-Sided Limit We write and call L the limit from the right (or right- hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.

8. 8 The Limit Thus we have a left-sided limit: And a right-sided limit: And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

9. 9 Example 1 f (x) = |x|/x at x = 0 The left and right limits are different, therefore there is no limit. 0

10. 10 Example 2 Example from my graphing calculator. The limit does not exist at 2! The limit exists at 4! 24 2 4

11. 11 Infinite Limits Sometimes as x approaches c, f (x) approaches infinity or negative infinity. Consider Notice as x approaches 2 what happens to the y values.

12. 12 Infinite Limits Be careful. Consider Notice as x approaches 2 what happens to the y values. The left limit is - ∞ while the right limit is + ∞. 12

13. 13 Limit Properties the limit of a constant is just the constant. the limit of a power is the power of the limit. the limit of the root of the root of the limit. (If n is even then c ≥ 0 )

14. 14 Limit Properties Continued Let f and g be two functions, and assume that the following two limits are real and exist. the limit of the sum or difference of the functions is equal to the sum of the limits. Then:

15. 15 Limit Properties Continued If: Then: the limit of the product of the functions is the product of the limits of the functions. the limit of the quotient of the functions is the quotient of the limits of the functions.

16. 16 Summary of Rules of Limits

17. 17 Example 1 x 2 – 3x = 2 2 – 3 · 2 =4 – 6 = -2. Many times we may find a limit by direct substitution.

18. 18

19. 19 Summary. We learned about left and right limits. We learned the definition of limit. We learned about limits of infinity. We learned about properties of limits. 19

20. 20 Summary from Text. If a function exist to the left and right of a given value the left and right limits will usually exist. Remember at a vertical asymptote the lift and right limits will be ± ∞. The graph can help us roughly determine limits. When left and right limits are not equal the limit does not exist. At vertical asymptotes left and right limits may be equal, hence a limit exist, or them may not be equal and then there is no limit. 20

21. 21 ASSIGNMENT §2.1; Page 36; 1 – 5. 21

22. Algebraic Limits Although graphs are often very useful in finding limits there are algebraic methods that are quick and accurate. Is one of those situations that comes to mind. However one must be careful in the case where In this situation algebraic techniques to remove the part of the denominator causing the problem are necessary 22

23. Indeterminate Form The term indeterminate is used because the limit may or may not exist. if and, then are said to be indeterminate. 23 If or

24. 24 Example 3 Be careful when a quotient is involved. However What happens at x = 2?

25. 25 Example 4 Actually, notice it does factor but this was so much more fun! 25 Notice that this one does not factor but we can be algebraically creative!

26. 26 Importance of Limits. Limits will be the building block of both of our major definitions in calculus and calculus has such very wide applications. The idea of a limit has always intrigued mathematicians. The idea of getting closer and closer to something and never reaching it has a certain Zen quality. Indeed you will find that the type of limit we always need to determine will be a 0/0 limit.

27. 27 Commentary This course is devoted to the development of the important concepts of the calculus, concepts that are so far reaching and that have exercised such an impact on the modern world that it is perhaps correct to say that without knowledge of them a person today can scarcely claim to be well educated. Howard Eves

28. 28 Summary. We learned that the indeterminate limit the 0/0 limit will become important to us. We learned about algebraic method for finding limits.

29. by Tony Carrillo

30. 30 ASSIGNMENT §2.2; Page 36; 1 – 10.