2008 W -C L Calculus(I)

(1)

Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

(2)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

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⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(3)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

n→∞

an=0

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(4)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

n→∞

an=0

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(5)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

n→∞

an=0

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(6)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

n→∞

an=0

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(7)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

n→∞

an=0

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(8)

Ch2: Limit and Continuity

Examles:

1 an = ¹_n

n→∞

an=0

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

1

2, 1, ¹₄, ¹₂, ¹₆,. . .

4 an = _n+1ⁿ ⇒an=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 :N → R n → f(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 = ¹_n

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(11)

Ch2.1: Limit of Sequence

Limit Laws

Definition

f :N → R n → f(n)

1 an = ¹_n

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(12)

Ch2.1: Limit of Sequence

Limit Laws

Definition

f :N → R n → f(n)

1 an = ¹_n

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(13)

Ch2.1: Limit of Sequence

Limit Laws

Definition

f :N → R n → f(n)

1 an = ¹_n

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(14)

Ch2.1: Limit of Sequence

Limit Laws

Definition

f :N → R n → f(n)

1 an = ¹_n

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(15)

Ch2.1: Limit of Sequence

Limit Laws

Definition

f :N → R n → f(n)

1 an = ¹_n

2 an = ⁽⁻¹⁾_nⁿ⁻¹

3 a2m = _m¹, a2m−1 = _2m¹

4 an = _n+1ⁿ ⇒an=1− _n+1¹

(16)

Note:

Consider the population model, n→ time t .

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

i.e. lim

n→∞

an

Calculate the limit of previous examples.

(17)

Note:

i.e. lim

n→∞

an

(18)

Note:

i.e. lim

n→∞

an

(19)

Note:

i.e. lim

n→∞

an

(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.

|an−a| < 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)

n→∞

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

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(22)

Definition (Limit)

n→∞

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

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(23)

Definition (Limit)

n→∞

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(24)

Definition (Limit)

n→∞

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(25)

Definition (Limit)

n→∞

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(26)

Definition (Limit)

n→∞

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(27)

Definition (Limit)

n→∞

Ex1: an = (−1)ⁿ

Ex2: an = 1

√n

(28)

Definition (Limit)

n→∞

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)

n→∞

anand lim

n→∞

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(31)

n→∞

anand lim

n→∞

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(32)

n→∞

anand lim

n→∞

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(33)

n→∞

anand lim

n→∞

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(34)

n→∞

anand lim

n→∞

nlim→∞

(an+bn) = lim

n→∞

an+ lim

n→∞

an lim

n→∞

bn nlim→∞

an

bn

=

nlim→∞

an nlim→∞

bn

if lim

n→∞

bn 6=0.

(35)

2008 W -C L Calculus(I)

Calculus (I)

Ch2: Limit and Continuity

Ch2: Limit and Continuity

Ch2: Limit and Continuity

Ch2: Limit and Continuity

Ch2: Limit and Continuity

Ch2: Limit and Continuity

Ch2: Limit and Continuity

Remark:

Ch2.1: Limit of Sequence

Limit Laws

Ch2.1: Limit of Sequence

Limit Laws

Ch2.1: Limit of Sequence

Limit Laws

Ch2.1: Limit of Sequence

Limit Laws

Ch2.1: Limit of Sequence

Limit Laws

Ch2.1: Limit of Sequence

Limit Laws

Thank you.