2. What is a limit? The precise definition of a limit is difficult to understand without first looking at some examples.

3. Suppose we want to calculate the area of a square. One way is to calculate the length of the side and square the result.

4. For example, if the length of the side is 4, then the area is 16. 4 Suppose we change this problem slightly.

5. Let’s say the length of the side is to 4. 4.1 SideArea 4.116.81 The area is close to 16. close

6. 4.01 SideArea 4.116.81 4.0116.0801 If the length of the side is to 4, is the area closer to 16? closer Let’s change this problem again.

7. What if we get even closer? 4.001 SideArea 4.116.81 4.0116.0801 4.00116.008001

8. And closer… 4.0001 SideArea 4.116.81 4.0116.0801 4.00116.008001 4.000116.00080001

9. And closer…… 4.00001 SideArea 4.116.81 4.0116.0801 4.00116.008001 4.000116.00080001 4.0000116.0000800001

10. What is a limit? From these calculations, we see that if the length of the side is very close to 4, then the area is very close to 16. Notice that as the length of the side gets closer and closer to 4, the area gets closer and closer to 16. Also notice, that as the area gets closer and closer to 16 the length of the sides get closer and closer to 4. SideArea 4.116.81 4.0116.0801 4.00116.008001 4.000116.00080001 4.0000116.0000800001

11. In this example we say that the limit of the area, as the side approaches 4 is 16. If we let x be the length of the side and f(x) be the area. Then the correct mathematical notation of this problem would be:

12. Consider the function Let’s look at another example We can make f(x) as close to 11 as we want by making x closer to 3. If x = 3, then f(x) = 11. What do you think f(x) would be if x is very close to 3? What do you think that x would be if f(x) is very close to 11?

13. What is a limit? We will use these three conditions as our first definition of a limit: The limit of f(x) as x approaches a is L. If x is close to a, then f(x) is close to L. As x gets closer and closer to a, then f(x) gets closer and closer to L. We can make f(x) as close to L as we want, by making x closer to a.

14. There is one more requirement we need to add to the definition of a limit consider this function: Let’s see what f(x) is, if x is very close to 1?

15. It appears that the limit as x approaches 1 is 2. xf(x) 1.12.1 1.012.01 1.0012.001 1.00012.0001 1.000012.00001

16. The problem here is to make sure that x does not equal 1 because the function is not defined there. We must require that as x gets closer to 1, it cannot equal 1. A similar restriction must be put on our definition of a limit.

17. What is a limit? If x is close to a, then f(x) is close to L? As x gets close to a but is not equal to a, then f(x) gets close to L. We can make f(x) as close to L as we want by making x close to a, but not equal to a. The limit of f(x) as x approaches a is L.

18. What is a limit? The limit of a function f(x) as x approaches a is L.

19. What is a limit? Remember the first example about the square?

20. Compare Notations

21. What is a limit? Remember the second example?

22. What is a limit? And the third?

23. Consider a factory that produces defense equipment. Assume that the production cost depends on a single item, t. So, let the production cost function be P(t) = 2t. Also, assume that the number of units produced is a function of t. Let the function for the number of units produced be U(t) = t2 + 2. The factory management is interested in knowing the cost per unit of the equipment produced in the factory. So, the cost per unit function, C(t), is obtained by dividing P(t) by U(t). The following table indicates the total production cost function, the number of units produced function, and the cost per unit function for various values of t.

24. tP(t)U(t)C(t) 1230.6666 10201020.2 100200100020.02 1000200010000020.002 10000200001000000020.0002 We can see from the table that as the value of t increases, the value of U(t) increases sharply when compared to the value of P(t). The increase in the value of the function U(t) cannot be compensated by the increase in the value of the function P(t). In other words, P(t) can be ignored for large values of t. Thus, we can say that as the value of t tends to infinity, the value of the function tends to 0.

25. To be continued……..

26. A. P. Calculus Lesson objectives: to obtain numerical evidence for the calculation of limits; to determine what appears to be the limit from the numerical evidence; and to become aware of some of the problems in using numerical evidence for the calculation of limits