1 / 34

3-1: Limits

3-1: Limits. Objectives: To evaluate limits numerically, graphically, and analytically. To evaluate infinite limits. f(x) = 3x +1. USING YOUR CALCULATORS, MAKE A TABLE OF VALUES TO FIND THE VALUE THAT f(x) IS APPROACHING AS x IS APPROACHING 1.

spencer
Download Presentation

3-1: Limits

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. f(x) = 3x +1 USING YOUR CALCULATORS, MAKE A TABLE OF VALUES TO FIND THE VALUE THAT f(x) IS APPROACHING AS x IS APPROACHING 1.

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

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

  10. 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)

  11. 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)

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

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

  14. 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

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

  16. Lets do some examples together, shall we???? Finding Limits Algebraically Worksheet--Classwork

  17. Examples: Evaluate the Limit

  18. What is the function’s value approaching as your x values get larger and larger? Smaller and smaller? Given:

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

  20. Rational Function Examples:

  21. WHEN WE ARE EVALUATING THESE LIMITS INVOLVING INFINITY, 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

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

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

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

  25. 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

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

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

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

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

More Related