## Calculus (I)

W^{EN}-C^{HING} L^{IEN}

Department of Mathematics National Cheng Kung University

2008

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

Consider the sequence 1,^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

Consider the sequence 1,^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

Consider the sequence 1,^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2: Limit and Continuity

Examles:

**1** *a**n* = ^{1}_{n}

^{1}_{2},^{1}_{3},^{1}_{4}, . . . ,_{n}^{1}, . . .
*As n*→ ∞*, a**n* →0

We say that the limit is zero. lim

*n*→∞

*a**n*=0

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

1

2, 1, ^{1}_{4}, ^{1}_{2}, ^{1}_{6},. . .

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

### Remark:

So, if

*n*lim→∞

*a**n*=*a,*

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

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

*f* :*N* → *R*
*n* → *f*(n)

*f is called a sequence.*

*a**n* =*f*(n), we write{*a**n*}to represent the entire sequence
*a*0,*a*1,*a*2, . . .

Examples:(On previous page)

**1** *a**n* = ^{1}_{n}

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

*f* :*N* → *R*
*n* → *f*(n)

*f is called a sequence.*

*a**n* =*f*(n), we write{*a**n*}to represent the entire sequence
*a*0,*a*1,*a*2, . . .

Examples:(On previous page)

**1** *a**n* = ^{1}_{n}

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

*f* :*N* → *R*
*n* → *f*(n)

*f is called a sequence.*

*a**n* =*f*(n), we write{*a**n*}to represent the entire sequence
*a*0,*a*1,*a*2, . . .

Examples:(On previous page)

**1** *a**n* = ^{1}_{n}

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

*f* :*N* → *R*
*n* → *f*(n)

*f is called a sequence.*

*a**n* =*f*(n), we write{*a**n*}to represent the entire sequence
*a*0,*a*1,*a*2, . . .

Examples:(On previous page)

**1** *a**n* = ^{1}_{n}

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

*f* :*N* → *R*
*n* → *f*(n)

*f is called a sequence.*

*a**n* =*f*(n), we write{*a**n*}to represent the entire sequence
*a*0,*a*1,*a*2, . . .

Examples:(On previous page)

**1** *a**n* = ^{1}_{n}

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

## Ch2.1: Limit of Sequence

### Limit Laws

Definition

*f* :*N* → *R*
*n* → *f*(n)

*f is called a sequence.*

*a**n* =*f*(n), we write{*a**n*}to represent the entire sequence
*a*0,*a*1,*a*2, . . .

Examples:(On previous page)

**1** *a**n* = ^{1}_{n}

**2** *a**n* = ^{(−1)}_{n}^{n}^{−}^{1}

**3** *a**2m* = _{m}^{1}*, a**2m*−1 = _{2m}^{1}

**4** *a**n* = _{n+1}* ^{n}* ⇒

*a*

*n*=1−

_{n+1}^{1}

Note:

Consider the population model,
*n*→ *time t*
.

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

i.e. lim

*n*→∞

*a**n*

Calculate the limit of previous examples.

Note:

Consider the population model,
*n*→ *time t*
.

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

i.e. lim

*n*→∞

*a**n*

Calculate the limit of previous examples.

Note:

Consider the population model,
*n*→ *time t*
.

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

i.e. lim

*n*→∞

*a**n*

Calculate the limit of previous examples.

Note:

Consider the population model,
*n*→ *time t*
.

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

i.e. lim

*n*→∞

*a**n*

Calculate the limit of previous examples.

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Definition (Limit)

The sequence {*a**n*}*has a limit a, lim*

*n*→∞

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

|*a**n*−*a*| < ǫ*, where n*>*N.*

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

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

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

*Ex1: a**n* = (−1)^{n}

*Ex2: a**n* = 1

√*n*

Theorem (limit laws)
*If lim*

*n*→∞

*a**n**and lim*

*n*→∞

*b**n* *exist and c is a constant, then*

*n*lim→∞

(a*n*+*b**n*) = lim

*n*→∞

*a**n*+ lim

*n*→∞

*b**n*
*n*lim→∞(ca*n*) = *c lim*

*n*→∞

*a**n*
*n*lim→∞(a*n**b**n*) = lim

*n*→∞

*a**n* lim

*n*→∞

*b**n*
*n*lim→∞

*a**n*

*b**n*

=

*n*lim→∞

*a**n*
*n*lim→∞

*b**n*

*if lim*

*n*→∞

*b**n* 6=*0.*

Theorem (limit laws)
*If lim*

*n*→∞

*a**n**and lim*

*n*→∞

*b**n* *exist and c is a constant, then*

*n*lim→∞

(a*n*+*b**n*) = lim

*n*→∞

*a**n*+ lim

*n*→∞

*b**n*
*n*lim→∞(ca*n*) = *c lim*

*n*→∞

*a**n*
*n*lim→∞(a*n**b**n*) = lim

*n*→∞

*a**n* lim

*n*→∞

*b**n*
*n*lim→∞

*a**n*

*b**n*

=

*n*lim→∞

*a**n*
*n*lim→∞

*b**n*

*if lim*

*n*→∞

*b**n* 6=*0.*

Theorem (limit laws)
*If lim*

*n*→∞

*a**n**and lim*

*n*→∞

*b**n* *exist and c is a constant, then*

*n*lim→∞

(a*n*+*b**n*) = lim

*n*→∞

*a**n*+ lim

*n*→∞

*b**n*
*n*lim→∞(ca*n*) = *c lim*

*n*→∞

*a**n*
*n*lim→∞(a*n**b**n*) = lim

*n*→∞

*a**n* lim

*n*→∞

*b**n*
*n*lim→∞

*a**n*

*b**n*

=

*n*lim→∞

*a**n*
*n*lim→∞

*b**n*

*if lim*

*n*→∞

*b**n* 6=*0.*

Theorem (limit laws)
*If lim*

*n*→∞

*a**n**and lim*

*n*→∞

*b**n* *exist and c is a constant, then*

*n*lim→∞

(a*n*+*b**n*) = lim

*n*→∞

*a**n*+ lim

*n*→∞

*b**n*
*n*lim→∞(ca*n*) = *c lim*

*n*→∞

*a**n*
*n*lim→∞(a*n**b**n*) = lim

*n*→∞

*a**n* lim

*n*→∞

*b**n*
*n*lim→∞

*a**n*

*b**n*

=

*n*lim→∞

*a**n*
*n*lim→∞

*b**n*

*if lim*

*n*→∞

*b**n* 6=*0.*

Theorem (limit laws)
*If lim*

*n*→∞

*a**n**and lim*

*n*→∞

*b**n* *exist and c is a constant, then*

*n*lim→∞

(a*n*+*b**n*) = lim

*n*→∞

*a**n*+ lim

*n*→∞

*b**n*
*n*lim→∞(ca*n*) = *c lim*

*n*→∞

*a**n*
*n*lim→∞(a*n**b**n*) = lim

*n*→∞

*a**n* lim

*n*→∞

*b**n*
*n*lim→∞

*a**n*

*b**n*

=

*n*lim→∞

*a**n*
*n*lim→∞

*b**n*

*if lim*

*n*→∞

*b**n* 6=*0.*

Theorem (limit laws)
*If lim*

*n*→∞

*a**n**and lim*

*n*→∞

*b**n* *exist and c is a constant, then*

*n*lim→∞

(a*n*+*b**n*) = lim

*n*→∞

*a**n*+ lim

*n*→∞

*b**n*
*n*lim→∞(ca*n*) = *c lim*

*n*→∞

*a**n*
*n*lim→∞(a*n**b**n*) = lim

*n*→∞

*a**n* lim

*n*→∞

*b**n*
*n*lim→∞

*a**n*

*b**n*

=

*n*lim→∞

*a**n*
*n*lim→∞

*b**n*

*if lim*

*n*→∞

*b**n* 6=*0.*