Chapter 1 Functions and Limits 1.4 Limit of a Function and Limit Laws

1. Chapter 1 Functions and Limits1.4 Limit of a Function and Limit Laws Many ideas of calculus originated with the following two geometric problems:

2. Traditionally, that portion of calculus arising from the tangent line problem is called differential calculus and that arising from the area problem is called integral calculus • Tangent lines and limits • Areas and limits • Decimals and limits*

3. Limits The most basic use of limits is to describe how a function behaves as the independent variable approaches a given value. For example, let us examine the behavior of the function for x values closer and closer to 2. We can see that the values of f(x) get closer and closer to 3 as values of x are selected closer and closer to 2 on either side of 2.

4. Limits We describe this by saying that the “limit of is 3 as x approaches 2 from either side”, and we write

5. Note: Since x is different from a, the value of f at a or even whether f is defined at a, has not bearing on the limit L Limits (An Informal View)

6. Ex: How does the function behave near x=1? Solution: The given formula defines f for all real numbers x except x=1. For any x≠1, f(x)=x+1. So by observing the graph, it is clear that we can make the value of f(x) as close as we want to 2 by choosing x close enough to 1.

7. We say that f(x) approaches the limit 2 as x approaches 1, and write

8. Example

9. Example Limit of the identity function Limit of the constant function

10. Example Discuss the behavior of the following functions as x0. It grows too large to have a limit It oscillates too much to have a limit It jumps

11. The Limit Laws

12. Examples

13. Example, Solution: Example: Solution:

14. Theorems

15. Examples For example, Solution: The function involved is a polynomial. So Example, Solution:

16. Eliminating Zero Denominators Algebraically When p(x)/q(x) is a rational function for which p (a) =0 and q(a)=0, the numerator and denominator must have one or more common factors of x – a. In this case, the limit of p(x)/q(x) as x  a can be found by canceling all common factors of x – a first. Here are some examples…

17. For example, Find Solution: Since 1 is a zero of both the numerator and the denominator, they share a common factor of x-1. The limit can be obtained as follows:

18. Example: Find Solution: The numerator and the denominator both have a zero at x=2, so there is common factor of x-2. Then

19. The Sandwich Theorem

20. Example Given that Find limxc u(x), no matter how complicated u is. Solution: Since The Sandwich Theorem implies that

21. Examples (a) Note that (b) Note that

22. Theorem

23. 1.5 The Precise Definition of a Limit

25. 1.6 One-sided Limits For example: consider the function

26. As x approaches 0 from the right, f(x) approaches 1, and similarly, as x approaches 0 from the left, f(x) approaches -1. We denote this by Here “+” indicates a limit from the right and “-” indicates a limit from the left.

27. One-Sided Limits

29. Example Example: Find

30. The relation between one-sided limits and two-sided limits In general, there is no guarantee that a function f will have a two-sided limit at a given point. In this case, we say that does not exist. Similarly for one-sided limits. Here we state the relation without formal proof

31. Ex: for the functions in the slide, find the one-sided and two sided limits at x=a if they exists.

32. Solution: • The functions in all three figures have the same one-sided limits as x->a, since the function are identical, except at x=a. These limits are • In all three cases the two-sided limit does not exist at x->a since the one-sided limits are not equal.

33. Example Example: for the function graphed below, find out for k=0, 2, 3, 4?

34. Limits Involving (sin / ) A central fact about sin /  is that in radian measure its limit as 0 is 1. We can see this from the figure below, and confirm it algebraically using the Sandwich Theorem.

35. Examples Find the following limits. Let =h/2.

36. Example:

37. Example:

38. 1.7 CONTINUITY Intuitively, the graph of a function can be described as a “continuous curve” if it has not breaks or holes. The graph of a function has a break or hole if any of the following conditions occur: • The function f is undefined at c • The limit of f(x) does not exist as x approaches c • The value of the function and the value of the limit at c are different.

40. Example Find the points at which the function f in below figure is continuous and the points at which f is not continuous. Why?

41. Continuity at a Point If a function is defined on an open interval containing c, except possibly at c itself, and f is not continuous at c, then we say that f is discontinuous at c.

42. Definitions A function f is right-continuous (continuous from the right) at a point x=2 in its domain if A function f is left-continuous (continuous from the left) at a point x=2 in its domain if

43. Continuity Test For one-sided continuity and continuity at an endpoint, the limits in part 2 And part 3 of the test should be replaced by the appropriate one-sided limits.

44. Example: Determine whether the following functions are continuous at x=-3. • Solution: • Observe that • f(x) is not continuous at x=-3 since it’s undefined at x=-3, • g(x) is not continuous at x=-3 since • h(x) is continuous

45. Example Example. The function y=|x| is discontinuous at every integer because The left –hand and right-hand limits are not equal as xn:

46. Discontinuities Below figure displays several common types of discontinuities:

47. Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. Example: (a) the function y=1/x is continuous function. (b )the function y=x and the constant functions are continuous everywhere.

48. Properties of Continuous Functions Algebraic combinations of continuous functions are continuous wherever they are defined.

49. Examples • Every polynomial is continuous everywhere. • A rational function is continuous at every point where the denominator is nonzero, and has discontinuities at the points where the denominator is zero. • The functions y=sinx and y=cosx are, in fact, continuous everywhere. It follows that all six trigonometric functions are then continuous wherever they are defined.

50. Composites All composites of continuous functions are continuous.