Lecture 2- BASIC LIMITS and INDETERMINATE FORM.pptx

Lesson about limits

1. Basic Limits and the Indeterminate Form

2. Definition of Limits If f(x) is a function and becomes arbitrarily close to a single number L as x approaches c from either side, then the limit of f(x) as x approaches c is L. This limit is written mathematically as and is read as “the limit of f(x) as x approaches c is L”  

3. Definition of Limits To illustrate this definition, let’s take for example the function whose graph is shown  

4. Definition of Limits For values other than x = 1, you can use standard curve-sketching techniques. At x = 1, however, it is not clear what to expect.

5. Definition of Limits To get an idea of the behavior of the graph of f(x) near x = 1, we can use two sets of x-values—one set that approaches 1 from the left and one set that approaches 1 from the right

6. Definition of Limits Hence, we could say that f(x) approaches the value 3 as the value of x approaches 1 from either the left or the right.

7. Definition of Limits Therefore, the limit of f(x) as x approaches 1 is 3.

8. Estimating a Limit Numerically

9. Estimating a Limit Numerically Example 1: For the function discuss the behavior of the values of f(x) when x gets closer to 2 using table  

10. Estimating a Limit Numerically Therefore,  

11. Estimating a Limit Numerically Example 2: Find the limit of as x approaches zero  

12. Estimating a Limit Numerically Example 2: Evaluate the function at several x-values near 0 and use the results to estimate the limit *As in the graph, we could see that f(0) is undefined. For this reason, we cannot find the limit by finding f(0) as in Example 1  

13. Estimating a Limit Numerically Example 2: To estimate the limit of f(x) as x approaches 0, a list of several values of x near zero from left and right would help.

14. Estimating a Limit Numerically Therefore,  

15. Estimating a Limit Numerically Example 3: For the function discuss the behavior of the values of f(x) when x is closer to 2. Does the limit exist?

16. Estimating a Limit Numerically * This function is not defined when x = 2. * The limit does not exist because the limit on the left and the limit on the right are not the same. represents the limit on the left of 2 represents the limit on the right of 2   x 1 1.9 1.99 2 2.001 2.01 2.1 2.5 f ( x ) -1 -1 -1 -1 ? 1 1 1 1

17. Estimating a Limit Numerically 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. 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. I n order for a limit to exist , the limit from the left and the limit from the right must exist and be equal.

18. Estimating a Limit Using Graph

19. Estimating a Limit Using Graph Example 1: From the given graph of f(x), answer the following f(0)          

20. Estimating a Limit Using Graph Example 1: From the given graph of f(x), answer the following f(1)          

21. Estimating a Limit Using Graph Example 1: From the given graph of f(x), answer the following f(3)          

22. Evaluating Limits Analytically

23. Example 1:   Try lim (x 4 + 3x – 2) X  -1 If you don’t get -4, try again

24. Example 2:   Property 8 Try lim X  -1 If you don’t get 2, try again

25. Example 3:   Note that this is a rational function with a nonzero denominator at x = -2 If you don’t get 1/3, try again

26. Example 4: If x < 5 If x > 5 Find: f(5)   This is called “Piecewise Function”

27. Example 4: If x < 5 If x > 5 Solution:  

28. Example 4: If x < 5 If x > 5 Solution:  

29. Example 1: Use algebraic and/or graphical techniques to analyze each of the following indeterminate forms  

30. Example 1: Solution  

31. Example 1: Solution  

32. Example 1: Solution  

33. Example 2: Evaluate the limit note: a 3 - b 3 = (a - b)(a 2 + ab + b 2 )  

34. Example 3: Find the limit  

35. Example 3: Find the limit  

36. SEATWORK:   2. 3.