Advanced Calculus (I)

Advanced Calculus (I) W EN -C HING L IEN Department of Mathematics National Cheng Kung University

W EN -C HING L IEN

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 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). x→a+

W EN -C HING L IEN

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 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). x→a+

W EN -C HING L IEN

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 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). x→a+

W EN -C HING L IEN

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 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). x→a+

W EN -C HING L IEN

Advanced Calculus (I)

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 f (x). x→a−

W EN -C HING L IEN

Advanced Calculus (I)

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 f (x). x→a−

W EN -C HING L IEN

Advanced Calculus (I)

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 f (x). x→a−

W EN -C HING L IEN

Advanced Calculus (I)

Theorem Let f be a real function. Then the limit lim f (x)

x→a

exists and equals L if and only if L = lim f (x) = lim f (x). x→a+

W EN -C HING L IEN

x→a−

Advanced Calculus (I)

Theorem Let f be a real function. Then the limit lim f (x)

x→a

exists and equals L if and only if L = lim f (x) = lim f (x). x→a+

W EN -C HING L IEN

x→a−

Advanced Calculus (I)

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 lim f (x) = L

x→∞

(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. W EN -C HING L IEN

Advanced Calculus (I)

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 lim f (x) = L

x→∞

(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. W EN -C HING L IEN

Advanced Calculus (I)

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 lim f (x) = L

x→∞

(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. W EN -C HING L IEN

Advanced Calculus (I)

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 lim f (x) = L

x→∞

(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. W EN -C HING L IEN

Advanced Calculus (I)

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 lim f (x) = L

x→∞

(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. W EN -C HING L IEN

Advanced Calculus (I)

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 lim f (x)

x→a x∈I

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

W EN -C HING L IEN

Advanced Calculus (I)

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 lim f (x)

x→a x∈I

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

W EN -C HING L IEN

Advanced Calculus (I)

Example: Prove that

2x 2 − 1 = −2. x→∞ 1 − x 2 lim

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Advanced Calculus (I)

Example: Prove that

2x 2 − 1 = −2. x→∞ 1 − x 2 lim

W EN -C HING L IEN

Advanced Calculus (I)

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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −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 1 expression above by 2 we have x 2 − 1/x 2 2x 2 − 1 = lim lim x→∞ −1 + 1/x 2 x→∞ 1 − x 2 limx→∞ (2 − 1/x 2 ) = limx→∞ (−1 + 1/x 2 ) 2 = −1

W EN -C HING L IEN

Advanced Calculus (I)

= −2.

2

Thank you.

W EN -C HING L IEN

Advanced Calculus (I)