Advanced Calculus (I)

WEN-CHING LIEN

Department of MathematicsNational Cheng Kung University

WEN-CHING LIEN Advanced Calculus (I)

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 afrom the right if and only if f is defined on some openinterval I with left endpoint a and for every ε > 0 there is aδ > 0 (which in general depends on ε, f, I, and a) such thata + δ ∈ I and

a < x < a + δ implies |f (x)− L| < ε.

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

f (a+) := L =: limx→a+

f (x).

WEN-CHING LIEN Advanced Calculus (I)

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 afrom the right if and only if f is defined on some openinterval I with left endpoint a and for every ε > 0 there is aδ > 0 (which in general depends on ε, f, I, and a) such thata + δ ∈ I and

a < x < a + δ implies |f (x)− L| < ε.

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

f (a+) := L =: limx→a+

f (x).

WEN-CHING LIEN Advanced Calculus (I)

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 afrom the right if and only if f is defined on some openinterval I with left endpoint a and for every ε > 0 there is aδ > 0 (which in general depends on ε, f, I, and a) such thata + δ ∈ I and

a < x < a + δ implies |f (x)− L| < ε.

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

f (a+) := L =: limx→a+

f (x).

WEN-CHING LIEN Advanced Calculus (I)

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 afrom the right if and only if f is defined on some openinterval I with left endpoint a and for every ε > 0 there is aδ > 0 (which in general depends on ε, f, I, and a) such thata + δ ∈ I and

a < x < a + δ implies |f (x)− L| < ε.

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

f (a+) := L =: limx→a+

f (x).

WEN-CHING LIEN Advanced Calculus (I)

Definition(ii)

A real function is said to converge to L as x approaches afrom the left if and only if f is defined on some openinterval I with right endpoint a and for every ε > 0 there isa δ > 0 (which in general depends on ε, f, I, and a) suchthat a− ε ∈ I and

a− δ < x < a implies |f (x)− L| < ε.

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

f (a−) := L =: limx→a−

f (x).

WEN-CHING LIEN Advanced Calculus (I)

Definition(ii)

A real function is said to converge to L as x approaches afrom the left if and only if f is defined on some openinterval I with right endpoint a and for every ε > 0 there isa δ > 0 (which in general depends on ε, f, I, and a) suchthat a− ε ∈ I and

a− δ < x < a implies |f (x)− L| < ε.

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

f (a−) := L =: limx→a−

f (x).

WEN-CHING LIEN Advanced Calculus (I)

Definition(ii)

A real function is said to converge to L as x approaches afrom the left if and only if f is defined on some openinterval I with right endpoint a and for every ε > 0 there isa δ > 0 (which in general depends on ε, f, I, and a) suchthat a− ε ∈ I and

a− δ < x < a implies |f (x)− L| < ε.

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

f (a−) := L =: limx→a−

f (x).

WEN-CHING LIEN Advanced Calculus (I)

TheoremLet f be a real function. Then the limit

limx→a

f (x)

exists and equals L if and only if

L = limx→a+

f (x) = limx→a−

f (x).

WEN-CHING LIEN Advanced Calculus (I)

TheoremLet f be a real function. Then the limit

limx→a

f (x)

exists and equals L if and only if

L = limx→a+

f (x) = limx→a−

f (x).

WEN-CHING LIEN Advanced Calculus (I)

Definition (2)(a)

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

|f (x)− L| < ε.

In this case, we write

limx→∞

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.

WEN-CHING LIEN Advanced Calculus (I)

Definition (2)(a)

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

|f (x)− L| < ε.

In this case, we write

limx→∞

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.

WEN-CHING LIEN Advanced Calculus (I)

Definition (2)(a)

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

|f (x)− L| < ε.

In this case, we write

limx→∞

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.

WEN-CHING LIEN Advanced Calculus (I)

Definition (2)(a)

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

|f (x)− L| < ε.

In this case, we write

limx→∞

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.

WEN-CHING LIEN Advanced Calculus (I)

Definition (2)(a)

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

|f (x)− L| < ε.

In this case, we write

limx→∞

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.

WEN-CHING LIEN Advanced Calculus (I)

TheoremLet a be an extended real number, and I be anondegenerate open interval which either contains a orhas a as one of its endpoints. Suppose further that f is areal function defined on I except possibly at a. Then

limx→ax∈I

f (x)

exists and equals L if and only if f (xn)→ L for allsequence xn ∈ I that satisfy xn 6= a and xn → a as n→∞.

WEN-CHING LIEN Advanced Calculus (I)

TheoremLet a be an extended real number, and I be anondegenerate open interval which either contains a orhas a as one of its endpoints. Suppose further that f is areal function defined on I except possibly at a. Then

limx→ax∈I

f (x)

exists and equals L if and only if f (xn)→ L for allsequence xn ∈ I that satisfy xn 6= a and xn → a as n→∞.

WEN-CHING LIEN Advanced Calculus (I)

Example:

Prove that

limx→∞

2x2 − 11− x2 = −2.

WEN-CHING LIEN Advanced Calculus (I)

Example:

Prove that

limx→∞

2x2 − 11− x2 = −2.

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Proof:

Since the limit of a product is the product of the limits, wehave by Example 3.15 that 1/xm → 0 as x →∞ for anym ∈ N. Multiplying numerator and denominator of the

expression above by1x2 we have

limx→∞

2x2 − 11− x2 = lim

x→∞

2− 1/x2

−1 + 1/x2

=limx→∞(2− 1/x2)

limx→∞(−1 + 1/x2)

=2−1

= −2. 2

WEN-CHING LIEN Advanced Calculus (I)

Thank you.

WEN-CHING LIEN Advanced Calculus (I)