3-1: Limits

1. 3-1: Limits Objectives: To evaluate limits numerically, graphically, and analytically. To evaluate infinite limits

2. USING YOUR CALCULATORS, MAKE A TABLE OF VALUES TO FIND THE VALUE THAT f(x) IS APPROACHING AS x IS APPROACHING 1 FROM THE LEFT AND FROM THE RIGHT. f(x) = 3x + 1 “As x is approaching 1 from the left” “As x is approaching 1 from the right”

3. 1. What do we know about the graph?2. What does the graph look like near x =1?

4. If the values of f(x) approach the number L as x approaches a from both the left and the right, we say that the limit L as xapproaches a exists and **You can use a table of values to find a limit by taking values of x very, very, very close to a on BOTH sides and see if they approach the same value Informal Definition of a Limit

5. A limit describes how the outputs of a function behave as the inputs approach some particular value. It is NOT necessarily the value of the function at that x value (but it could be). WHAT???????????????? Yes, this is true  Something weird….

6. RIGHT-HAND LIMIT (RHL) (The limit as x approaches a from the right) LEFT-HAND LIMIT(LHL) (The limit as x approaches a from the left) One-Sided Limits

7. IN ORDER FOR A LIMIT TO EXIST, THE FUNCTION HAS TO BE APPROACHING THE SAMEVALUE FROM BOTH THE LEFT AND THE RIGHT (LHL = RHL) =

8. Let’s take a look at limits graphically!!

9. Example continues…

12. Graph the following function. Then find the limit.

13. Look at a table of values and the graph of What happens as x approaches 2? DOES NOT EXIST

14. is not a number. It is used to describe a situation where something increases or decreases without bound (gets more and more negative or more and more positive)

15. A LIMIT DOES NOT EXIST (DNE) WHEN: 1. The RHL and LHL as x approaches some value a are BOTH or BOTH - . We write or , but the limit DNE. • The RHL as x approaches some value a is and the LHL as x approaches the same value is - or vice versa. • LHL ≠ RHL (The fancy dancy explanations are on page 154)

17. Find all the zeros: 2x3+x2-x PRIZE ROUND

18. Sum/Difference Rule: • Product Rule: • Constant Multiple Rule: • Quotient Rule: • Power Rule: Properties of LimitsIf L, M, a and k are real numbers and and , then

19. If one of the limits for one of the functions DNE when using the properties, then the limit for the combined function DNE. Note:

20. 1. If p(x) is a polynomial, then 2. , where c is a constant 3. Other important properties and limits…..

21. Take a look at p. 165 # 25 and 30.

22. Try substitution (If a is in the domain of the function this works). If you get 0/0 when you substitute, there is something you can do to simplify!! • If substitution doesn’t work, simplify, if possible. Then evaluate limit. • Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by the conjugate. Simplify and evaluate. You can always use a table or a graph to reinforce your conclusion How to Find Limits Algebraically

23. Factor the following: 1. x3-27 • 8x3+1 • 4x2-9 Prize---Prize---Prize

24. Lets do some examples together, shall we???? Handout—Finding Limits Algebraically—Classwork I do #1,3,5,8,10,11 with you You try #2,4,7,9 Finding Limits Algebraically Worksheet--Classwork

25. Evaluate the limit: Some trickier examples

26. Evaluate the limit:

27. Examples: Evaluate the Limit

28. What is the function’s value approaching as the x values get larger and larger in the positive direction? Larger and larger in the negative direction? Given:

29. FOR ANY POSITIVE REAL NUMBER n AND ANY REAL NUMBER c : and TO FIND THE FOR ANY RATIONAL FUNCTION , DIVIDE NUMERATOR AND DENOMINATOR BY THE VARIABLE EXPRESSION WITH THE LARGEST POWER IN DENOMINATOR.

30. Rational Function Examples:

31. WHEN WE ARE EVALUATING THESE LIMITS AS x ±∞, WHAT ESSENTIALLY ARE WE FINDING? • WE LEARNED IT IN PRE-CALC WHEN WE GRAPHED RATIONAL FUNCTIONS • WHAT DOES THE END-BEHAVIOR OF A FUNCTION TELL US? • IT BEGINS WITH AN “H” Prize

32. DEFINTION OF HORIZONTAL ASYMPTOTE • THE LINE y=b IS A HORIZONTAL ASYMPTOTE OF THE GRAPH OF y=f(x) IF EITHER OR This is Calculus!!!! Woohoo!!

33. Examples: a.)Evaluate the Limitb.) What is the equation for the HA?

34. Extra Examples, if needed.

35. ***NOTE: A function can have more than one horizontal asymptote. Take a look at these graphs.

36. If the degree of the numerator is less than the degree of the denominator, the limit of the rational function is 0. • If the degree of the numerator is = to the degree of the denominator, the limit of the rational function is the ratio of the leading coefficients. • If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function approaches ±∞. Guidelines for finding limits as x ±∞ of Rational Functions

38. LOOK AT THE GRAPH OF VERTICAL ASYMPTOTES AND INFINITE LIMITS

39. The line x=a is a vertical asymptote of y=f(x) if either: OR Vertical Asymptote: Definition

40. Sum/difference: • Product: • Quotient: Properties of Infinite LimitsLet c and L be real numbers and let f and g be functions such that

41. 3. Evaluate the limit: Find the vertical asymptote. Prove using a limit.