Lesson 1 - Introduction to Limits.pptx

Lesson 1 on Introduction to Limits an informal and more intuitive approach.

1. Introduction to Limits An Informal Approach

2. Example 1 – Finding a rectangle of Maximum Area You are given 24 inches of wire and are asked to form a rectangle whose area is as large as possible. What dimensions should the rectangle have?

3. Example 1 – Solution Let w represent the width of the rectangle, and let l represent the length of the rectangle. Because 2 w + 2 l = 24, it follows that l = 12 – w So, the area of the rectangle is A = lw A = (12 – w ) w A = 12w – w 2 . Using this model for area, you can experiment with different values of w to see how to obtain the maximum area.

4. Example 1 – Solution After trying several values, it appears that the maximum area occurs when w = 6 as shown in the table. In limit terminology, you can say that “the limit of A as w approaches 6 is 36.” This is written as

5. What is a limit? What are some examples of limits?

6. Limits – An Informal Approach Up to now you have used algebra and geometry to solve problems where things are basically staying the same or changing at the same rate. Calculus is a branch of mathematics that deals with things that are changing. Volume of an expanding balloon Acceleration of rocket ship that is changing every part of a second The rate of change at any point on a curve

7. Limits – An Informal Approach One of the most fundamental ideas of calculus is limits. Limits allow us to look at what happens in a very, very small region around a point. Two of the major formal definitions of calculus depend on limits. What is a limit?

8. Example 2 – Limit of a Function Consider the function, whose domain is the set of all numbers except Although cannot be evaluated because substituting for x results in the undefined quantity 0/0 But, f ( x ) can be calculated at any number x that is very close to  

9. Example 2 – Limit of a Function As x approaches from either the left or right, the function values f(x) appear to be approaching 8. In other words, when x is near , f(x) is near 8 .  

10. Example 2 – Limit of a Function To interpret the numerical information graphically , We say 8 is the limit of f(x) as x approaches -4.

11. Limit of a Function – An Informal Approach Suppose L denotes a finite number. The notation of f ( x ) approaching L as x approaches a number a can be defined informally in the following manner. Informal Definition If f ( x ) can be made arbitrarily close to the number L by taking x sufficiently close to but different from the number a , from both the left and right sides of a , then the limit of f ( x ) as x approaches a is L .

12. Notation The discussion of the limit concept is facilitated by using a special notation. If we let the arrow symbol → represent the word approach , then the symbolism indicates that x approaches a number a from the left , that is, through numbers that are less than a , and signifies that x approaches a from the right , that is, through numbers that are greater than a . signifies that x approaches a from both sides , in other words, from the left and the right sides of a on a number line.  

13. Example 3 – Estimating a Limit Numerically Use a table to estimate

14. Example 3 – Estimating a Limit Numerically USING A TABLE TO ESTIMATE A LIMIT. Solution: Let f ( x ) = 3 x – 2. Construct a table that shows values of f ( x ) for two sets of x -values—one set that approaches 2 from the left and one that approaches 2 from the right. From the table, it appears that the closer x gets to 2, the closer f ( x ) gets to 4. So, you can estimate the limit to be 4.

15. Example 3 – Estimating a Limit Numerically The graph adds further support to this conclusion,

16. Example 4 – Estimating a Limit Graphically Use a graph to estimate

17. Example 4 – Estimating a Limit Numerically Reinforce with the graph. f(x) has a limit as x  0 even though the function is not defined at x = 0. .

18. One-sided Limits In general, if a function f(x) can be made arbitrarily close to a number L 1 by taking x sufficiently close to, but not equal to, a number a from the left, then we write The number L 1 is said to be the left-hand limit of f(x) as x approaches a . Similarly, if f(x) can be made arbitrarily close to a number L 2 by taking x sufficiently close to, but not equal to, a number a from the right, then L 2 is the right-hand limit of f(x) as x approaches a , and we write

19. Two-sided Limits If both the left-hand limit and the right hand limit exists and have a common value L, Then we say that L is the limit of f(x) as x approaches a and write This limit is said to be a two-sided limit .

20. Two-sided Limits  

21. Existence and Nonexistence The existence of a limit of a function f as x approaches a (from one side or both sides) does NOT depend on whether f is defined at a but ONLY on whether f is defined for x near the number a .

22. Existence and Nonexistence For example, if the previous function is modified in the following manner Then, f(-4) is defined, and f(-4)= 5, but still, the limit is equal to 8.

23. Existence and Nonexistence

24. Limits that Fails to Exist Limits that Exist

25. Limits that Fails to Exist

26. Example 5 – Comparing Left and Right Behavior Show that the limit does not exist by analyzing the graph. 1.) 2.) 3.)

27. Introduction to Limits – Extended Still THE informal approach...

28. Based on the graph above, tell whether the limit is true or false.

29. A Limit That Exists The graph of the function is shown. As seen from the graph and the accompanying tables, it seems plausible that And consequently,  

30. A Limit That Exists The graph of the piecewise-defined function is given below.

31. A Limit That Does Not Exists The graph of the piecewise-defined function is given below.

32. A Limit That Does Not Exists The graph of the greatest integer function or floor function