1. . Blast from the Past Find point(s) of intersection 1. 2. 3.4. 5.

2. . Blast from the Past Find point(s) of intersection 1. 2. 3.4. 5.

3. . Blast from the Past In exercises 1 – 4 write the inequality in the form 1. 2. 3. 4. In exercises 5 and 6 write the fraction in reduced form 5. 6.

4. . Blast from the Past In exercises 1 – 4 write the inequality in the form 1. 2. 3. 4. In exercises 5 and 6 write the fraction in reduced form 5. 6.

5. Finding Limits Graphically and Numerically Limits are restrictions…mathematically f(x) is restricted by the graph of the function. In this section you will estimate a limit using a numerical or graphical approach. Learn different ways that a limit can fail to exist.

6. Lets analyze the graph of f (x):. To get an idea of the behavior of the graph of f (x) near x = 0 lets use a set of values approaching 0 from the left and set of values approaching 0 from the right. Estimating a Limit Numerically

7. Slide 2- 7 Analyzing the table we see although x cannot equal 0,we can come arbitrarily close to 0 and as a result the function f(x) moves arbitrarily close to 2. x-0.01-0.001-0.0001-0.0000100.000010.00010.0010.01 f(x)1.99491.999501.99995 2.000052.00052.004992.0049 Which leads to the limit notation: Which reads as “the limit of f(x) as x approaches 0 is 2

8. . Exploration: Estimate numerically the x0.750.90.990.99911.0011.011.11.25 f(x)2.3132.7102.9702.997?3.0033.0303.3103.813 X1.751.91.991.99922.0012.012.12.25 f(x)????????? Complete the chart and estimate numerically the

9. Finding a Limit Graphically Use the graph of f(x) to determine each limit:

10. Behavior That Differs from Right to Left: The following graphs of f(x) represent functions that the limit of f(x) as x, approaches a, does not exist. As x approaches a from the right: Limits That Fail to Exist As x approaches a from the left: Because f(x) approaches a different number from the left side of a than it approaches from the right side, the limit does not exist.

11. Unbounded Behavior: The following graph of f(x) represent function that the does not exist Oscillating Behavior: The following graph of f(x) represent function that the does not exist

12. Exit Ticket 1.Write a brief description of the meaning of the notation 2.If, can you conclude anything about the limit of f(x) as x approaches 2? Explain your reasoning.

13. . Blast from the Past In exercises 1 and 2 rationalize the numerator 1. 2. In exercise 3 simplify expression 3. 4. If then

14. . Blast from the Past In exercises 1 and 2 rationalize the numerator 1. 2. In exercise 3 simplify expression 3. 4. If then

15. 5. Use the graph to find the limit (if it exists). If the limit does not exist, explain why.

16. . 6. Use the graph to find the limit (if it exists). If the limit does not exist, explain why.

17. 7. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not explain why. a. b. c. d.

18. 8. Use the graph of the function f to decide whether the value of the given quantity exists. If it does, find it. If not explain why. a. b. c. d. e. f.

19. Evaluating Limits Analytically In this section you will evaluate a limit using properties of limits. Develop a strategy for finding limits. Evaluate a limit using dividing out and rationalizing techniques..

20. Previous section we learned that does not depend on the value of the function at x = c. However, it may happen a limit may be precisely f(c), the limit can be evaluated by Direct Substitution. These well-behaved functions are continuous. Analytical Approach

21. Determine the limits of the functions Remember a limit may be precisely f(c) and the limit can be evaluated by Direct Substitution if the function is continuous at x = c.

22. Properties of Limits: Where f(x) and g(x) are functions with limits that exists such thatand 1. Scalar multiple: where b is a constant 2.Sum or difference: 3.Product: 4.Quotient: 5.Power:

23. Indeterminate Form of a Limit Direct Substitution may produce the solution 0/0, an indeterminate form Dividing Technique Rationalizing Technique Simplifying Technique

24. Exploration In exercises 1 – 5 evaluate the limits 1. 2. 3. 4. 5.

25. Limits of Trigonometric Functions Let a be a real number in the domain of the given trigonometric function. Two Special Trigonometric Limits 1. 2.

26. Exploration of Limits of Functions The principle of direct substitution can be expanded to include exponential, logarithmic, piece-wise, and inverse trigonometric functions 1. 5. 2.6. 3.7. 4.8.

27. . Exit Ticket 1.Letfind 2.Let find 3.What is meant by indeterminate form?