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# 2008 W -C L Calculus(I)

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## Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

2008

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## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(3)

## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(4)

## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(5)

## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(6)

## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(7)

## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(8)

## Ch2: Limit and Continuity

Examles:

1 an = 1n

Consider the sequence 1,12,13,14, . . . ,n1, . . . As n→ ∞, an →0

We say that the limit is zero. lim

n→∞

an=0

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

1

2, 1, 14, 12, 16,. . .

4 an = n+1nan=1− n+11

(9)

### Remark:

So, if

nlim→∞

an=a,

it means that a can be described by an infinite sequence of approximations.

(10)

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

f :NR nf(n)

f is called a sequence.

an =f(n), we write{an}to represent the entire sequence a0,a1,a2, . . .

Examples:(On previous page)

1 an = 1n

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

4 an = n+1nan=1− n+11

(11)

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

f :NR nf(n)

f is called a sequence.

an =f(n), we write{an}to represent the entire sequence a0,a1,a2, . . .

Examples:(On previous page)

1 an = 1n

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

4 an = n+1nan=1− n+11

(12)

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

f :NR nf(n)

f is called a sequence.

an =f(n), we write{an}to represent the entire sequence a0,a1,a2, . . .

Examples:(On previous page)

1 an = 1n

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

4 an = n+1nan=1− n+11

(13)

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

f :NR nf(n)

f is called a sequence.

an =f(n), we write{an}to represent the entire sequence a0,a1,a2, . . .

Examples:(On previous page)

1 an = 1n

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

4 an = n+1nan=1− n+11

(14)

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

f :NR nf(n)

f is called a sequence.

an =f(n), we write{an}to represent the entire sequence a0,a1,a2, . . .

Examples:(On previous page)

1 an = 1n

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

4 an = n+1nan=1− n+11

(15)

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

f :NR nf(n)

f is called a sequence.

an =f(n), we write{an}to represent the entire sequence a0,a1,a2, . . .

Examples:(On previous page)

1 an = 1n

2 an = (−1)nn1

3 a2m = m1, a2m1 = 2m1

4 an = n+1nan=1− n+11

(16)

Note:

Consider the population model, ntime t .

We are interested in the ”long-time” behavior.

i.e. lim

n→∞

an

Calculate the limit of previous examples.

(17)

Note:

Consider the population model, ntime t .

We are interested in the ”long-time” behavior.

i.e. lim

n→∞

an

Calculate the limit of previous examples.

(18)

Note:

Consider the population model, ntime t .

We are interested in the ”long-time” behavior.

i.e. lim

n→∞

an

Calculate the limit of previous examples.

(19)

Note:

Consider the population model, ntime t .

We are interested in the ”long-time” behavior.

i.e. lim

n→∞

an

Calculate the limit of previous examples.

(20)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(21)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =aif for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(22)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0,there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(23)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(24)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(25)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(26)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(27)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(28)

Definition (Limit)

The sequence {an}has a limit a, lim

n→∞

an =a if for every ǫ >0, there exist an integer N s.t.

|ana| < ǫ, where n>N.

If the limit exists, the sequence is called ”convergent”.

(or an converges to a as n tends to infinity.)

If the sequence has no limit, it is called ”divergent”.

Ex1: an = (−1)n

Ex2: an = 1

n

(29)

Theorem (limit laws) If lim

n→∞

anand lim

n→∞

bn exist and c is a constant, then

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

bn nlim→∞(can) = c lim

n→∞

an nlim→∞(anbn) = lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(30)

Theorem (limit laws) If lim

n→∞

anand lim

n→∞

bn exist and c is a constant, then

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

bn nlim→∞(can) = c lim

n→∞

an nlim→∞(anbn) = lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(31)

Theorem (limit laws) If lim

n→∞

anand lim

n→∞

bn exist and c is a constant, then

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

bn nlim→∞(can) = c lim

n→∞

an nlim→∞(anbn) = lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(32)

Theorem (limit laws) If lim

n→∞

anand lim

n→∞

bn exist and c is a constant, then

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

bn nlim→∞(can) = c lim

n→∞

an nlim→∞(anbn) = lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(33)

Theorem (limit laws) If lim

n→∞

anand lim

n→∞

bn exist and c is a constant, then

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

bn nlim→∞(can) = c lim

n→∞

an nlim→∞(anbn) = lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(34)

Theorem (limit laws) If lim

n→∞

anand lim

n→∞

bn exist and c is a constant, then

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

bn nlim→∞(can) = c lim

n→∞

an nlim→∞(anbn) = lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(35)

## Thank you.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

W EN -C HING L IEN Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung