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

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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

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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

(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

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Remark:

So, if

nlim→∞

an=a,

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

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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

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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

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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

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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

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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.

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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.

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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.

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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.

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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

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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

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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

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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

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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.

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Thank you.

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