Limits of FunctionsMATH 464/506, Real Analysis

J. Robert Buchanan

Department of Mathematics

Summer 2007

J. Robert Buchanan Limits of Functions

Cluster Points

Definition

Let A ⊆ R. A point c ∈ R is a cluster point of A if for everyδ > 0 there exists at least one point of x ∈ A, x 6= c such that|x − c| < δ.

Remarks:

This is equivalent to “A point c ∈ R is a cluster point of A ifevery δ-neighborhood Vδ(c) of c contains at least one pointof A distinct from c.

Point c does not have to be a point in A.

J. Robert Buchanan Limits of Functions

Cluster Points

Definition

Let A ⊆ R. A point c ∈ R is a cluster point of A if for everyδ > 0 there exists at least one point of x ∈ A, x 6= c such that|x − c| < δ.

Remarks:

This is equivalent to “A point c ∈ R is a cluster point of A ifevery δ-neighborhood Vδ(c) of c contains at least one pointof A distinct from c.

Point c does not have to be a point in A.

J. Robert Buchanan Limits of Functions

Cluster Points

Definition

Let A ⊆ R. A point c ∈ R is a cluster point of A if for everyδ > 0 there exists at least one point of x ∈ A, x 6= c such that|x − c| < δ.

Remarks:

This is equivalent to “A point c ∈ R is a cluster point of A ifevery δ-neighborhood Vδ(c) of c contains at least one pointof A distinct from c.

Point c does not have to be a point in A.

J. Robert Buchanan Limits of Functions

Cluster Points (cont.)

Theorem

A number c ∈ R is a cluster point of a subset A of R if and onlyif there exists a sequence (an) in A such that lim(an) = c andan 6= c for all n ∈ N.

Proof.

J. Robert Buchanan Limits of Functions

Cluster Points (cont.)

Theorem

A number c ∈ R is a cluster point of a subset A of R if and onlyif there exists a sequence (an) in A such that lim(an) = c andan 6= c for all n ∈ N.

Proof.

J. Robert Buchanan Limits of Functions

Examples

Example

Set Cluster Points(0, 1) [0, 1]

{1, 2, . . . , n} ∅N ∅

{1n : n ∈ N} {0}[0, 1] ∩ Q [0, 1]

J. Robert Buchanan Limits of Functions

Examples

Example

Set Cluster Points(0, 1) [0, 1]

{1, 2, . . . , n} ∅N ∅

{1n : n ∈ N} {0}[0, 1] ∩ Q [0, 1]

J. Robert Buchanan Limits of Functions

Examples

Example

Set Cluster Points(0, 1) [0, 1]

{1, 2, . . . , n} ∅N ∅

{1n : n ∈ N} {0}[0, 1] ∩ Q [0, 1]

J. Robert Buchanan Limits of Functions

Examples

Example

Set Cluster Points(0, 1) [0, 1]

{1, 2, . . . , n} ∅N ∅

{1n : n ∈ N} {0}[0, 1] ∩ Q [0, 1]

J. Robert Buchanan Limits of Functions

Examples

Example

Set Cluster Points(0, 1) [0, 1]

{1, 2, . . . , n} ∅N ∅

{1n : n ∈ N} {0}[0, 1] ∩ Q [0, 1]

J. Robert Buchanan Limits of Functions

Examples

Example

Set Cluster Points(0, 1) [0, 1]

{1, 2, . . . , n} ∅N ∅

{1n : n ∈ N} {0}[0, 1] ∩ Q [0, 1]

J. Robert Buchanan Limits of Functions

Definition of the Limit

Definition

Let A ⊆ R and let c be a cluster point of A. Suppose f : A → R

then a real number L is said to be a limit of f at c if, given anyǫ > 0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ,then |f (x) − L| < ǫ.

Remarks:

Notation: limx→c

f (x) = L

δ usually depends on ǫ, therefore δ ≡ δ(ǫ)

0 < |x − c| < δ implies x 6= c

If the limit of f at c does not exist we say f diverges at c.

J. Robert Buchanan Limits of Functions

Definition of the Limit

Definition

Let A ⊆ R and let c be a cluster point of A. Suppose f : A → R

then a real number L is said to be a limit of f at c if, given anyǫ > 0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ,then |f (x) − L| < ǫ.

Remarks:

Notation: limx→c

f (x) = L

δ usually depends on ǫ, therefore δ ≡ δ(ǫ)

0 < |x − c| < δ implies x 6= c

If the limit of f at c does not exist we say f diverges at c.

J. Robert Buchanan Limits of Functions

Definition of the Limit

Definition

Let A ⊆ R and let c be a cluster point of A. Suppose f : A → R

then a real number L is said to be a limit of f at c if, given anyǫ > 0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ,then |f (x) − L| < ǫ.

Remarks:

Notation: limx→c

f (x) = L

δ usually depends on ǫ, therefore δ ≡ δ(ǫ)

0 < |x − c| < δ implies x 6= c

If the limit of f at c does not exist we say f diverges at c.

J. Robert Buchanan Limits of Functions

Definition of the Limit

Definition

Let A ⊆ R and let c be a cluster point of A. Suppose f : A → R

then a real number L is said to be a limit of f at c if, given anyǫ > 0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ,then |f (x) − L| < ǫ.

Remarks:

Notation: limx→c

f (x) = L

δ usually depends on ǫ, therefore δ ≡ δ(ǫ)

0 < |x − c| < δ implies x 6= c

If the limit of f at c does not exist we say f diverges at c.

J. Robert Buchanan Limits of Functions

Definition of the Limit

Definition

Let A ⊆ R and let c be a cluster point of A. Suppose f : A → R

then a real number L is said to be a limit of f at c if, given anyǫ > 0 there exists a δ > 0 such that if x ∈ A and 0 < |x − c| < δ,then |f (x) − L| < ǫ.

Remarks:

Notation: limx→c

f (x) = L

δ usually depends on ǫ, therefore δ ≡ δ(ǫ)

0 < |x − c| < δ implies x 6= c

If the limit of f at c does not exist we say f diverges at c.

J. Robert Buchanan Limits of Functions

Uniqueness of Limits

Theorem

If f : A → R and if c is a cluster point of A, then f can have onlyone limit at c.

Proof.

Theorem

Let f : A → R and let c be a cluster point of A. Then thefollowing statements are equivalent.

1 limx→c

f (x) = L

2 Given any ǫ-neighborhood Vǫ(L) of L, there exists aδ-neighborhood Vδ(c) of c such that if x 6= c is any point inVδ(c) ∩ A, then f (x) ∈ Vǫ(L).

Proof.J. Robert Buchanan Limits of Functions

Uniqueness of Limits

Theorem

If f : A → R and if c is a cluster point of A, then f can have onlyone limit at c.

Proof.

Theorem

Let f : A → R and let c be a cluster point of A. Then thefollowing statements are equivalent.

1 limx→c

f (x) = L

2 Given any ǫ-neighborhood Vǫ(L) of L, there exists aδ-neighborhood Vδ(c) of c such that if x 6= c is any point inVδ(c) ∩ A, then f (x) ∈ Vǫ(L).

Proof.J. Robert Buchanan Limits of Functions

Uniqueness of Limits

Theorem

If f : A → R and if c is a cluster point of A, then f can have onlyone limit at c.

Proof.

Theorem

Let f : A → R and let c be a cluster point of A. Then thefollowing statements are equivalent.

1 limx→c

f (x) = L

2 Given any ǫ-neighborhood Vǫ(L) of L, there exists aδ-neighborhood Vδ(c) of c such that if x 6= c is any point inVδ(c) ∩ A, then f (x) ∈ Vǫ(L).

Proof.J. Robert Buchanan Limits of Functions

Uniqueness of Limits

Theorem

If f : A → R and if c is a cluster point of A, then f can have onlyone limit at c.

Proof.

Theorem

Let f : A → R and let c be a cluster point of A. Then thefollowing statements are equivalent.

1 limx→c

f (x) = L

2 Given any ǫ-neighborhood Vǫ(L) of L, there exists aδ-neighborhood Vδ(c) of c such that if x 6= c is any point inVδ(c) ∩ A, then f (x) ∈ Vǫ(L).

Proof.J. Robert Buchanan Limits of Functions

ǫ-δ Game

How to prove limx→c

f (x) = L:

1 Let ǫ > 0.2 Find a value of δ > 0 that will guarantee that whenever x is

within a distance δ from c (but not equal to c), f (x) is withina distance ǫ of L.

3 Prove that for this value of δ,

∀x ∈ D(f ), 0 < |x − c| < δ ⇒ |f (x) − L| < ǫ.

J. Robert Buchanan Limits of Functions

Examples

Example

1 limx→c

b = b

2 limx→c

x = c

3 limx→c

x2 = c2

4 limx→c

1x

=1c

if c > 0.

5 limx→2

x3 − 4x2 + 1

=45

J. Robert Buchanan Limits of Functions

Sequential Criterion for Limits

Theorem (Sequential Criterion)

Let f : A → R and let c be a cluster point of A. Then thefollowing are equivalent.

1 limx→c

f (x) = L.

2 For every sequence (xn) in A that converges to c such thatxn 6= c for all n ∈ N, the sequence (f (xn)) converges to L.

Proof.

J. Robert Buchanan Limits of Functions

Sequential Criterion for Limits

Theorem (Sequential Criterion)

Let f : A → R and let c be a cluster point of A. Then thefollowing are equivalent.

1 limx→c

f (x) = L.

2 For every sequence (xn) in A that converges to c such thatxn 6= c for all n ∈ N, the sequence (f (xn)) converges to L.

Proof.

J. Robert Buchanan Limits of Functions

Divergence Criteria

Theorem

Let A ⊆ R, let f : A → R and let c ∈ R be a cluster point of A.1 If L ∈ R, then f does not have limit L at c if and only if there

exists a sequence (xn) in A with xn 6= c for all n ∈ N suchthat the sequence (xn) converges to c but the sequence(f (xn)) does not converge to L.

2 The function f does not have a limit at c if and only if thereexists a sequence (xn) in A with xn 6= c for all n ∈ N suchthat the sequence (xn) converges to c but the sequence(f (xn)) does not converge in R.

J. Robert Buchanan Limits of Functions

Divergence Examples

Example

1 limx→0

1x

2 limx→0

+1 if x > 0,0 if x = 0,−1 if x < 0.

3 limx→0

sin(

1x

)

J. Robert Buchanan Limits of Functions

Homework

Read Section 4.1

Page 104: 1, 3 , 7, 9, 12 , 14

Boxed problems should be written up separately and submittedfor grading at class time on Friday.

J. Robert Buchanan Limits of Functions