Introduction to Limits

Transcription

1 for An Introduction to Calculus Pramudya 1 1 Senior Lecturer in Department of Industrial Computing Faculty of Information and Communication Technology Universiti Teknikal Malaysia Melaka with LaTEX February 16, 2014

2 Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

3 why where Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

4 why where You should be abble to... find the limit of linear and quadratic functions by using the intuitive meaning of limit (C2); find the limit of improper rational functions by using the intuitive meaning of limit (C2); compare the limit of a function with the numbers produced by calculators or computers (C2) (optional); prove the limit of linear functions by the use of precise definition of limit (C3) (optional); use the main limit theorem to find the limits of at least 3 (three) complex functions (C3);

5 why where You should be abble to... find the limit of linear and quadratic functions by using the intuitive meaning of limit (C2); find the limit of improper rational functions by using the intuitive meaning of limit (C2); compare the limit of a function with the numbers produced by calculators or computers (C2) (optional); prove the limit of linear functions by the use of precise definition of limit (C3) (optional); use the main limit theorem to find the limits of at least 3 (three) complex functions (C3);

6 why where You should be abble to... find the limit of linear and quadratic functions by using the intuitive meaning of limit (C2); find the limit of improper rational functions by using the intuitive meaning of limit (C2); compare the limit of a function with the numbers produced by calculators or computers (C2) (optional); prove the limit of linear functions by the use of precise definition of limit (C3) (optional); use the main limit theorem to find the limits of at least 3 (three) complex functions (C3);

7 why where You should be abble to... find the limit of linear and quadratic functions by using the intuitive meaning of limit (C2); find the limit of improper rational functions by using the intuitive meaning of limit (C2); compare the limit of a function with the numbers produced by calculators or computers (C2) (optional); prove the limit of linear functions by the use of precise definition of limit (C3) (optional); use the main limit theorem to find the limits of at least 3 (three) complex functions (C3);

8 why where You should be abble to... find the limit of linear and quadratic functions by using the intuitive meaning of limit (C2); find the limit of improper rational functions by using the intuitive meaning of limit (C2); compare the limit of a function with the numbers produced by calculators or computers (C2) (optional); prove the limit of linear functions by the use of precise definition of limit (C3) (optional); use the main limit theorem to find the limits of at least 3 (three) complex functions (C3);

9 why where continued... determine whether a step function is continuous or not at a point, given the graph of the function (C3). prove whether a rational function is continue or not at a point (C3);

10 why where continued... determine whether a step function is continuous or not at a point, given the graph of the function (C3). prove whether a rational function is continue or not at a point (C3);

11 why where Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

12 why where One of resources... Varberg, D., Purcell,E.J, and Rigdon, S.E.(2007) Calculus, 9th Ed. Pearson International Edition, Pearson-Education

13 Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

14 Intuitive Definition To say that lim x c f (x) = L means that when x near but different from c then f (c) is near L. Example 01: Find lim(4x 5) x 3 Solution: When x is near 3, 4x 5 is near = 7. We write lim(4x 5) = 7 x 3

15 Intuitive Definition To say that lim x c f (x) = L means that when x near but different from c then f (c) is near L. Example 01: Find lim(4x 5) x 3 Solution: When x is near 3, 4x 5 is near = 7. We write lim(4x 5) = 7 x 3

16 Intuitive Definition To say that lim x c f (x) = L means that when x near but different from c then f (c) is near L. Example 01: Find lim(4x 5) x 3 Solution: When x is near 3, 4x 5 is near = 7. We write lim(4x 5) = 7 x 3

17 Contd. Example 02: Find x 2 x 6 lim x 3 x 3. Solution: To get an idea of is happening as x approaches 3, we could use a calculator to evaluate the given expression, for example, at 3.1, 3.01, and so on. But it much more precise to use algebra. x 2 x 6 (x 3)(x + 2) lim = lim x 3 x 3 x 3 x 3 = lim(x + 2) = = 5 x 3 Why is the cancellation of x 3 legitimate?

18 Contd. Example 02: Find x 2 x 6 lim x 3 x 3. Solution: To get an idea of is happening as x approaches 3, we could use a calculator to evaluate the given expression, for example, at 3.1, 3.01, and so on. But it much more precise to use algebra. x 2 x 6 (x 3)(x + 2) lim = lim x 3 x 3 x 3 x 3 = lim(x + 2) = = 5 x 3 Why is the cancellation of x 3 legitimate?

19 Contd. Example 02: Find x 2 x 6 lim x 3 x 3. Solution: To get an idea of is happening as x approaches 3, we could use a calculator to evaluate the given expression, for example, at 3.1, 3.01, and so on. But it much more precise to use algebra. x 2 x 6 (x 3)(x + 2) lim = lim x 3 x 3 x 3 x 3 = lim(x + 2) = = 5 x 3 Why is the cancellation of x 3 legitimate?

20 Calculators may fool you Example 04: Find. lim x 0 [x2 cos x 10, 000 ] Solution: Trying to plot it or using a table over [ 1, 1], you will have a misleading message as the limit looks like 0. Using a delicate way, we have lim x 0 [x2 cos x 10, 0000 ] = , 000 = 1 10, 000

21 Calculators may fool you Example 04: Find. lim x 0 [x2 cos x 10, 000 ] Solution: Trying to plot it or using a table over [ 1, 1], you will have a misleading message as the limit looks like 0. Using a delicate way, we have lim x 0 [x2 cos x 10, 0000 ] = , 000 = 1 10, 000

22 Calculators may fool you Example 04: Find. lim x 0 [x2 cos x 10, 000 ] Solution: Trying to plot it or using a table over [ 1, 1], you will have a misleading message as the limit looks like 0. Using a delicate way, we have lim x 0 [x2 cos x 10, 0000 ] = , 000 = 1 10, 000

23 Precise Meaning of Limit To say that lim x c f (x) = L means that for each given ϸ > 0 (no matter how small) there is a corresponding δ > 0 such that f (x) L < ϸ, provided that 0 < x c < δ; that is. Example 05: Prove that 0 < x c < δ = f (x) L < ϸ lim(3x 7) = 5 x 4 The proof consists of preliminary analysis and formal proof.

24 Precise Meaning of Limit To say that lim x c f (x) = L means that for each given ϸ > 0 (no matter how small) there is a corresponding δ > 0 such that f (x) L < ϸ, provided that 0 < x c < δ; that is. Example 05: Prove that 0 < x c < δ = f (x) L < ϸ lim(3x 7) = 5 x 4 The proof consists of preliminary analysis and formal proof.

25 Precise Meaning of Limit To say that lim x c f (x) = L means that for each given ϸ > 0 (no matter how small) there is a corresponding δ > 0 such that f (x) L < ϸ, provided that 0 < x c < δ; that is. Example 05: Prove that 0 < x c < δ = f (x) L < ϸ lim(3x 7) = 5 x 4 The proof consists of preliminary analysis and formal proof.

26 Preliminary analysis Let ϸ be any positive number. We must produce a δ > 0 such that 0 < x 4 < δ = (3x 7) 5 < ϸ Consider the inequality on the right. (3x 7) 5 < ϸ 3x 12 < ϸ 3(x 4) < ϸ 3 (x 4) < ϸ x 4 < ϸ 3 Now we see how to choose δ; that is, δ = ϸ/3. Of course, any smaller δ would work.

27 Preliminary analysis Let ϸ be any positive number. We must produce a δ > 0 such that 0 < x 4 < δ = (3x 7) 5 < ϸ Consider the inequality on the right. (3x 7) 5 < ϸ 3x 12 < ϸ 3(x 4) < ϸ 3 (x 4) < ϸ x 4 < ϸ 3 Now we see how to choose δ; that is, δ = ϸ/3. Of course, any smaller δ would work.

28 Preliminary analysis Let ϸ be any positive number. We must produce a δ > 0 such that 0 < x 4 < δ = (3x 7) 5 < ϸ Consider the inequality on the right. (3x 7) 5 < ϸ 3x 12 < ϸ 3(x 4) < ϸ 3 (x 4) < ϸ x 4 < ϸ 3 Now we see how to choose δ; that is, δ = ϸ/3. Of course, any smaller δ would work.

29 Formal proof Let ϸ > 0 be given. Choose δ = ϸ/3 Then 0 < x 4 < δ implies that (3x 7) 5 = 3x 12 = 3(x 4) = 3 (x 4) < 3δ = ϸ If you read this chain of equalities and an equality from left to right and use the transitive properties of = and <, you have that (3x 7) 5 < ϸ

30 Formal proof Let ϸ > 0 be given. Choose δ = ϸ/3 Then 0 < x 4 < δ implies that (3x 7) 5 = 3x 12 = 3(x 4) = 3 (x 4) < 3δ = ϸ If you read this chain of equalities and an equality from left to right and use the transitive properties of = and <, you have that (3x 7) 5 < ϸ

31 Formal proof Let ϸ > 0 be given. Choose δ = ϸ/3 Then 0 < x 4 < δ implies that (3x 7) 5 = 3x 12 = 3(x 4) = 3 (x 4) < 3δ = ϸ If you read this chain of equalities and an equality from left to right and use the transitive properties of = and <, you have that (3x 7) 5 < ϸ

32 Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

33 List of main theorems Let n be a positive integer, k be a constant, and f and g be functions that have limits at c. Then 1 lim x c k = k; 2 lim x c x = c; 3 lim x c kf (x) = k lim x c f (x); 4 lim[f (x) + g(x)] = lim f (x) + lim g(x); x c x c x c 5 lim[f (x) g(x)] = lim f (x) lim g(x); x c x c x c 6 lim x c [f (x).g(x)] = lim x c f (x). lim x c g(x);

34 contd. f (x) 1 lim x c g(x) = lim f (x) x c lim x c g(x) provided that lim x c g(x) 0; 2 lim[f (x)] n = [lim f (x)] n ; x c x c 3 lim n f (x) = n lim f (x), provided lim f (x) > 0 when n is even. x c x c x c

35 Application of the Theorems Example 06: Find lim x 4 x x Solution: lim x 4 x x = lim x x 4 lim x = x 4 lim x x 4 4 = 1 lim x lim 9 = 1 x 4 x 4 4 = = 5 4 [lim x 4 x] 2 + 9

36 Application of the Theorems Example 06: Find lim x 4 x x Solution: lim x 4 x x = lim x x 4 lim x = x 4 lim x x 4 4 = 1 lim x lim 9 = 1 x 4 x 4 4 = = 5 4 [lim x 4 x] 2 + 9

37 Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

38 Continuity at a Point Let f be defined at an open interval containing c. We say that f is continuous at c iff lim f (x) = f (c) x c The definition requires three things, which are 1 lim x c f (x) exits; 2 f (c) exits; 3 lim x c f (x) = f (c).

39 Continuity at a Point Let f be defined at an open interval containing c. We say that f is continuous at c iff lim f (x) = f (c) x c The definition requires three things, which are 1 lim x c f (x) exits; 2 f (c) exits; 3 lim x c f (x) = f (c).

40 Examples Function f (x) = x 2 9 is continuous at x = 3; Function g(x) = x2 9 x 3 is not continuous (discontinuous) at x = 3; Function g(x) = x2 9 x 3 can be made to be continuous at x = 3 by defining g(3) = 6

41 Examples Function f (x) = x 2 9 is continuous at x = 3; Function g(x) = x2 9 x 3 is not continuous (discontinuous) at x = 3; Function g(x) = x2 9 x 3 can be made to be continuous at x = 3 by defining g(3) = 6

42 Examples Function f (x) = x 2 9 is continuous at x = 3; Function g(x) = x2 9 x 3 is not continuous (discontinuous) at x = 3; Function g(x) = x2 9 x 3 can be made to be continuous at x = 3 by defining g(3) = 6

43 Outline 1 OBJECTIVES RESOURCES 2 LIMIT OF A FUNCTION LIMIT THEOREMS CONTINUITY

44 Exercise 1 Find the indicated limits by using intuitive meaning of limit. 1 lim x 3 (x 5) 2 lim t 1 (1 2t) 3 lim x 2 (x2 + 2t 1)

45 Exercise 2 1 lim t 1 (t 2 1) 2 t lim t 7 t+7 (t+4)(t 2) 4 3 lim t 2 (3t 6) 2

46 Exercise 3 Use calculators or computers to find the following limits. Plot the function near the limit point. 1 sin x lim x π 2x 2 1 cos t lim t 0 2t