## Advanced Calculus (I)

WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

## 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 ).

## 3.2 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 ).

## 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 ).

## 3.2 One-sided Limits And Limits At Infinty

Definition (1)
Let a ∈**R.**

(i)

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 ).

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 ).

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 ).

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 ).

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 ).

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 ).

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.

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

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.

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

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.

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 → ∞.

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 → ∞.

**Example:**

Prove that

x →∞lim

2x^{2}− 1

1 − x^{2} = −2.

**Example:**

Prove that

x →∞lim

2x^{2}− 1

1 − x^{2} = −2.

**Proof:**

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

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2

**Proof:**

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

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2

**Proof:**

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

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2

**Proof:**

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

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2

**Proof:**

^{m} → 0 as x → ∞ for any
m ∈**N. Multiplying numerator and denominator of the**
expression above by 1

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2

**Proof:**

^{m} → 0 as x → ∞ for any
m ∈**N. Multiplying numerator and denominator of the**
expression above by 1

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2

**Proof:**

^{m} → 0 as x → ∞ for any
m ∈**N. Multiplying numerator and denominator of the**
expression above by 1

x^{2} we have

x →∞lim

2x^{2}− 1

1 − x^{2} = lim

x →∞

2 − 1/x^{2}

−1 + 1/x^{2}

= limx →∞(2 − 1/x^{2})
limx →∞(−1 + 1/x^{2})

= 2

−1 = −2. 2