• 沒有找到結果。

N/A
N/A
Protected

Copied!
27
0
0

(1)

WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

(2)

## 3.2 One-sided Limits And Limits At Infinty

Definition (1) Let a ∈R.

(i)

A real function is said to converge to L as x approaches a from the right if and only if f is defined on some open interval I with left endpoint a and for every  > 0 there is a δ >0 (which in general depends on , f, I, and a) such that a + δ ∈ I and

a < x < a + δ implies |f (x ) − L| < .

In this case we call L the right-hand limit of f at a, and denote it by

f (a+) := L =: lim f (x ).

(3)

## 3.2 One-sided Limits And Limits At Infinty

Definition (1) Let a ∈R.

(i)

A real function is said to converge to L as x approaches a from the right if and only if f is defined on some open interval I with left endpoint a and for every  > 0 there is a δ >0 (which in general depends on , f, I, and a) such that a + δ ∈ I and

a < x < a + δ implies |f (x ) − L| < .

In this case we call L the right-hand limit of f at a, and denote it by

f (a+) := L =: lim

x →a+f (x ).

(4)

## 3.2 One-sided Limits And Limits At Infinty

Definition (1) Let a ∈R.

(i)

A real function is said to converge to L as x approaches a from the right if and only if f is defined on some open interval I with left endpoint a and for every  > 0 there is a δ >0 (which in general depends on , f, I, and a) such that a + δ ∈ I and

a < x < a + δ implies |f (x ) − L| < .

In this case we call L the right-hand limit of f at a, and denote it by

f (a+) := L =: lim f (x ).

(5)

## 3.2 One-sided Limits And Limits At Infinty

Definition (1) Let a ∈R.

(i)

A real function is said to converge to L as x approaches a from the right if and only if f is defined on some open interval I with left endpoint a and for every  > 0 there is a δ >0 (which in general depends on , f, I, and a) such that a + δ ∈ I and

a < x < a + δ implies |f (x ) − L| < .

In this case we call L the right-hand limit of f at a, and denote it by

f (a+) := L =: lim

x →a+f (x ).

(6)

Definition (ii)

A real function is said to converge to L as x approaches a from the left if and only if f is defined on some open interval I with right endpoint a and for every  > 0 there is a δ > 0 (which in general depends on , f, I, and a) such that a −  ∈ I and

a − δ < x < a implies |f (x ) − L| < .

In this case we call L the left-hand limit of f at a and denote it by

f (a−) := L =: lim

x →a−f (x ).

(7)

Definition (ii)

A real function is said to converge to L as x approaches a from the left if and only if f is defined on some open interval I with right endpoint a and for every  > 0 there is a δ > 0 (which in general depends on , f, I, and a) such that a −  ∈ I and

a − δ < x < a implies |f (x ) − L| < .

In this case we call L the left-hand limit of f at a and denote it by

f (a−) := L =: lim

x →a−f (x ).

(8)

Definition (ii)

A real function is said to converge to L as x approaches a from the left if and only if f is defined on some open interval I with right endpoint a and for every  > 0 there is a δ > 0 (which in general depends on , f, I, and a) such that a −  ∈ I and

a − δ < x < a implies |f (x ) − L| < .

In this case we call L the left-hand limit of f at a and denote it by

f (a−) := L =: lim

x →a−f (x ).

(9)

Theorem

Let f be a real function. Then the limit

x →alimf (x )

exists and equals L if and only if L = lim

x →a+f (x ) = lim

x →a−f (x ).

(10)

Theorem

Let f be a real function. Then the limit

x →alimf (x )

exists and equals L if and only if L = lim

x →a+f (x ) = lim

x →a−f (x ).

(11)

Definition (2) (a)

f (x ) → L as x → ∞ if and only if for any given  > 0, there is an M ∈R such that for x > M,

|f (x) − L| < .

In this case, we write

x →∞lim f (x ) = L (b)

f (x ) → +∞ as x → a if and only if for any given M ∈R, there is a δ > 0 such that

f (x ) > M for 0 < |x − a| < δ.

(12)

Definition (2) (a)

f (x ) → L as x → ∞ if and only if for any given  > 0, there is an M ∈R such that for x > M,

|f (x) − L| < .

In this case, we write

x →∞lim f (x ) = L (b)

f (x ) → +∞ as x → a if and only if for any given M ∈R, there is a δ > 0 such that

(13)

Definition (2) (a)

f (x ) → L as x → ∞ if and only if for any given  > 0, there is an M ∈R such that for x > M,

|f (x) − L| < .

In this case, we write

x →∞lim f (x ) = L (b)

f (x ) → +∞ as x → a if and only if for any given M ∈R, there is a δ > 0 such that

f (x ) > M for 0 < |x − a| < δ.

(14)

Definition (2) (a)

f (x ) → L as x → ∞ if and only if for any given  > 0, there is an M ∈R such that for x > M,

|f (x) − L| < .

In this case, we write

x →∞lim f (x ) = L (b)

f (x ) → +∞ as x → a if and only if for any given M ∈R, there is a δ > 0 such that

(15)

Definition (2) (a)

f (x ) → L as x → ∞ if and only if for any given  > 0, there is an M ∈R such that for x > M,

|f (x) − L| < .

In this case, we write

x →∞lim f (x ) = L (b)

f (x ) → +∞ as x → a if and only if for any given M ∈R, there is a δ > 0 such that

f (x ) > M for 0 < |x − a| < δ.

(16)

Theorem

Let a be an extended real number, and I be a

nondegenerate open interval which either contains a or has a as one of its endpoints. Suppose further that f is a real function defined on I except possibly at a. Then

x →alim

x ∈I

f (x )

exists and equals L if and only if f (xn) →L for all

sequence xn ∈ I that satisfy xn6= a and xn→ a as n → ∞.

(17)

Theorem

Let a be an extended real number, and I be a

nondegenerate open interval which either contains a or has a as one of its endpoints. Suppose further that f is a real function defined on I except possibly at a. Then

x →alim

x ∈I

f (x )

exists and equals L if and only if f (xn) →L for all

sequence xn ∈ I that satisfy xn6= a and xn→ a as n → ∞.

(18)

Prove that

x →∞lim

2x2− 1

1 − x2 = −2.

(19)

Prove that

x →∞lim

2x2− 1

1 − x2 = −2.

(20)

### Proof:

Since the limit of a product is the product of the limits, we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(21)

### Proof:

Since the limit of a product is the product of the limits,we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(22)

### Proof:

Since the limit of a product is the product of the limits, we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(23)

### Proof:

Since the limit of a product is the product of the limits, we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(24)

### Proof:

Since the limit of a product is the product of the limits, we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(25)

### Proof:

Since the limit of a product is the product of the limits, we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(26)

### Proof:

Since the limit of a product is the product of the limits, we have by Example 3.15 that 1/xm → 0 as x → ∞ for any m ∈N. Multiplying numerator and denominator of the expression above by 1

x2 we have

x →∞lim

2x2− 1

1 − x2 = lim

x →∞

2 − 1/x2

−1 + 1/x2

= limx →∞(2 − 1/x2) limx →∞(−1 + 1/x2)

= 2

−1 = −2. 2

(27)

## Thank you.

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

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung