1. 2-1: RATES OF CHANGE AND LIMITS Objectives: To evaluate limits numerically, graphically, and analytically. To use properties of limits

2. Average Speed or Rate of Change Distance Covered Elapsed time A rock breaks free from the top of a cliff. What is the average speed during the first 2 seconds?? (y=16t 2 )

3. What if I wanted to know the speed at EXACTLY 2 seconds (use same function)? Let us use t = 2 and t= 2 + h.

4. Definition of a Limit Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement Means “The values f(x) of the function f approach or equal L as the values of x approach (but do not necessarily equal) c.

5. In other words… 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 x approaches a exists and **Please note..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.

6. Evaluating numerically…using a table of values. Evaluate Try: x-.999-.9999-1.0001-1.001 f(x)

7. One-Sided Limits RIGHT-HAND LIMIT (RHL) (The limit of f as x approaches c from the right) LEFT-HAND LIMIT(LHL) (The limit of f as x approaches c from the left)

8. IN ORDER FOR A LIMIT TO EXIST, THE FUNCTION HAS TO BE APPROACHING THE SAME VALUE FROM BOTH THE LEFT AND THE RIGHT (LHL and RHL must exist and be equal) IF = THEN

12. a.) Graph the function b.) Determine the LHL and the RHL c.) Does the limit exist? Explain.

13. Properties of Limits: If L, M, c and k are real numbers and and then: 1. Sum and Difference rule: 2. Product Rule: 3. Constant Multiple Rule: 4. Quotient Rule: 5. Power Rule:

14. Evaluating Algebraically Theorems: Polynomial and Rational Functions 1. If f(x) = a n x n + a n-1 x n-1 +…+a 0 is any polynomial function and c is a real number, then SUBSTITUTE!!!!!! 2.If f(x) and g(x) are polynomials and c is a real number, then SUBSTITUTE!!!

15. EXAMPLES

16. PRIZE ROUND Factor: 1. x 3 +1 2. 8x 3 -27 3. t 2 +5t-6 4. 25x 2 -64

17. To evaluate limits algebraically: 1. Try substitution. (c has to be in the domain). If you get 0/0, there is something you can do!! 2. If substitution doesn’t work, factor if possible, simplify, then try to evaluate 3. Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by conjugate. Simplify and evaluate. 4. Use table or graph to reinforce your conclustion.

18. Examples: Evaluate the limit.

19. Evaluate the limit:

20. Evaluate.

21. Sandwich (Squeeze) Theorem If g(x) < f(x) < h(x) when x is near c (except possibly at c) and THEN

22. Show

23. Useful Limits to Know!! Evaluate… 1. 2. 3.

24. Evaluate 1. 2. 3.