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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