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 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 ).
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
2x2− 1
1 − x2 = −2.
Example:
Prove that
x →∞lim
2x2− 1
1 − x2 = −2.
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
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
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
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
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
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
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
