## Advanced Calculus (I)

WEN-CHING LIEN

Department of Mathematics National Cheng Kung University

WEN-CHINGLIEN **Advanced Calculus (I)**

## 3.1 Two-Sided Limits

Definition

*Let a*∈**R, Let I be an open interval that contains a, and**
let f be a real function defined everywhere on I except
*possibly at a. Then f*(x)*is said to converge to L, as x*
*approaches a, if and only if for every*ǫ >0 there is aδ >0
(which in general depends on ǫ, f, I and a) such that

0< |*x* −*a*| < δ *implies* |*f*(x) −*L*| < ǫ.

In this case we write

*L*= lim*f*(x)

## 3.1 Two-Sided Limits

Definition

*Let a*∈**R, Let I be an open interval that contains a, and**
let f be a real function defined everywhere on I except
*possibly at a. Then f*(x)*is said to converge to L, as x*
*approaches a, if and only if for every*ǫ >0 there is aδ >0
(which in general depends on ǫ, f, I and a) such that

0< |*x* −*a*| < δ *implies* |*f*(x) −*L*| < ǫ.

In this case we write

*L*= lim

*x*→*a**f*(x)

*and call L the limit of f*(x)as x approaches a.

WEN-CHINGLIEN **Advanced Calculus (I)**

## 3.1 Two-Sided Limits

Definition

*Let a*∈**R, Let I be an open interval that contains a, and**
let f be a real function defined everywhere on I except
*possibly at a. Then f*(x)*is said to converge to L, as x*
*approaches a, if and only if for every*ǫ >0 there is aδ >0
(which in general depends on ǫ, f, I and a) such that

0< |*x* −*a*| < δ *implies* |*f*(x) −*L*| < ǫ.

In this case we write

*L*= lim*f*(x)

**Example:**

*1. f*(x) =3, lim

*x→1**f*(x) =?

*2. f*(x) =*3x, lim*

*x→1**f*(x) =?

*3. f*(x) =*x*^{2}, lim

*x→1**f*(x) =?

*4. f*(x) =√
*x, lim*

*x→1**f*(x) =?

WEN-CHINGLIEN **Advanced Calculus (I)**

**Example:**

*1. f*(x) =3, lim

*x→1**f*(x) =?

*2. f*(x) =*3x, lim*

*x→1**f*(x) =?

*3. f*(x) =*x*^{2}, lim

*x→1**f*(x) =?

*4. f*(x) =√
*x, lim*

*x→1**f*(x) =?

**Example:**

*1. f*(x) =3, lim

*x→1**f*(x) =?

*2. f*(x) =*3x, lim*

*x→1**f*(x) =?

*3. f*(x) =*x*^{2}, lim

*x→1**f*(x) =?

*4. f*(x) =√
*x, lim*

*x→1**f*(x) =?

WEN-CHINGLIEN **Advanced Calculus (I)**

**Example:**

*1. f*(x) =3, lim

*x→1**f*(x) =?

*2. f*(x) =*3x, lim*

*x→1**f*(x) =?

*3. f*(x) =*x*^{2}, lim

*x→1**f*(x) =?

*4. f*(x) =√
*x, lim*

*x→1**f*(x) =?

**Example:**

*1. f*(x) =3, lim

*x→1**f*(x) =?

*2. f*(x) =*3x, lim*

*x→1**f*(x) =?

*3. f*(x) =*x*^{2}, lim

*x→1**f*(x) =?

*4. f*(x) =√
*x, lim*

*x→1**f*(x) =?

WEN-CHINGLIEN **Advanced Calculus (I)**

**Remark:**

*Let a*∈**R, let I be an open interval that contains a, and let**
f,g be real functions defined everywhere on I except

*possibly at a. If f*(x) = *g(x*)*for all x* ∈*I*\ {*a*}*and f*(x) →*L*
*as x* →*a, then g(x)also has a limit as x* →*a, and*

*x→a*lim*g(x*) = lim

*x→a**f*(x).

**Remark:**

*Let a*∈**R, let I be an open interval that contains a, and let**
f,g be real functions defined everywhere on I except

*possibly at a. If f*(x) = *g(x*)*for all x* ∈*I*\ {*a*}*and f*(x) →*L*
*as x* →*a, then g(x)also has a limit as x* →*a, and*

*x→a*lim*g(x*) = lim

*x→a**f*(x).

WEN-CHINGLIEN **Advanced Calculus (I)**

**Example:**

*g(x*) = *x*^{3}+*x*^{2}−*x*−1
*x*^{2}−1 , lim

*x*→1*g(x*) =?

**Example:**

*g(x*) = *x*^{3}+*x*^{2}−*x*−1
*x*^{2}−1 , lim

*x*→1*g(x*) =?

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Sequential Characterization of Limits)

*Let a*∈**R, let I be an open interval that contains a, and let***f be a real function defined everywhere on I except*

*possibly at a. Then*

*L*= lim

*x→a**f*(x)

*exists if and only if f*(x*n*) →*L as n*→ ∞*for every*
*sequence x**n* ∈*I*\ {*a*}*that converges to a as n*→ ∞*.*

Theorem (Sequential Characterization of Limits)

*Let a*∈**R, let I be an open interval that contains a, and let***f be a real function defined everywhere on I except*

*possibly at a. Then*

*L*= lim

*x→a**f*(x)

*exists if and only if f*(x*n*) →*L as n*→ ∞*for every*
*sequence x**n* ∈*I*\ {*a*}*that converges to a as n*→ ∞*.*

WEN-CHINGLIEN **Advanced Calculus (I)**

**Example:**

Prove that

*f*(x) =
(

sin(1

*x*) *x* 6=0

0 *x* =0

*has no limit as x* →0.

**Example:**

Prove that

*f*(x) =
(

sin(1

*x*) *x* 6=0

0 *x* =0

*has no limit as x* →0.

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1),we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

Clearly, *both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

Clearly, *both a**n**and b**n* *converge to 0 as n*→ ∞. On the
other hand, *Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
other hand, *Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
3.6,*the limit of f(x), as x* →0, cannot exist. 2

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

WEN-CHINGLIEN **Advanced Calculus (I)**

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
3.6,*the limit of f(x), as x* →0, cannot exist. 2

**Proof:**

*By examing the graph of y* =*f*(x)(see Figure 3.1), we are
led to consider two extremes:

*a**n*:= 2

(4n+1)π *and b**n* := 2

(4n+3)π, *n* ∈**N.**

*Clearly, both a**n**and b**n* *converge to 0 as n*→ ∞. On the
*other hand, Since f*(a*n*) =*1 and f*(b*n*) = −*1 for all n* ∈**N,**
*f*(a*n*) →*1 and f*(b*n*) → −*1 as n* → ∞. Thus by Theorem
*3.6, the limit of f(x), as x* →0, cannot exist. 2

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Suppose that a*∈ **R, that I is an open interval that contains***a, and that f,g are real functions defined everywhere on I*
*except possibly at a. If f(x) and g(x) converge as x*

*approaches a, then so do* (f +*g)(x*),(fg)(x),(αf)(x), and
(f/g)(x)*(when the limit of g(x) is nonzero).*

*In fact,*

*x→a*lim(f +*g)(x) =* lim

*x→a**f*(x) + lim

*x→a**g(x*),

*x*lim→*a*(αf)(x) = αlim

*x*→*a**f*(x),

*x→a*lim(fg)(x) = lim

*x→a**f*(x)lim

*x→a**g(x),*
*and (when the limit of g(x) is nonzero)*

Theorem

*Suppose that a*∈ **R, that I is an open interval that contains***a, and that f,g are real functions defined everywhere on I*
*except possibly at a.* *If f(x) and g(x) converge as x*

*approaches a, then so do* (f +*g)(x*),(fg)(x),(αf)(x), and
(f/g)(x)*(when the limit of g(x) is nonzero).*

*In fact,*

*x→a*lim(f +*g)(x) =* lim

*x→a**f*(x) + lim

*x→a**g(x*),

*x*lim→*a*(αf)(x) = αlim

*x*→*a**f*(x),

*x→a*lim(fg)(x) = lim

*x→a**f*(x)lim

*x→a**g(x),*
*and (when the limit of g(x) is nonzero)*

*x→a*lim

*f*
*g*

(x) = lim*x→a**f*(x)
lim*x*→*a**g(x*).

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Suppose that a*∈ **R, that I is an open interval that contains***a, and that f,g are real functions defined everywhere on I*
*except possibly at a. If f(x) and g(x) converge as x*

*approaches a, then so do* (f +*g)(x*),(fg)(x),(αf)(x), and
(f/g)(x)*(when the limit of g(x) is nonzero).*

*In fact,*

*x→a*lim(f +*g)(x) =* lim

*x→a**f*(x) + lim

*x→a**g(x*),

*x*lim→*a*(αf)(x) = αlim

*x*→*a**f*(x),

*x→a*lim(fg)(x) = lim

*x→a**f*(x)lim

*x→a**g(x),*
*and (when the limit of g(x) is nonzero)*

Theorem

*Suppose that a*∈ **R, that I is an open interval that contains***a, and that f,g are real functions defined everywhere on I*
*except possibly at a. If f(x) and g(x) converge as x*

*approaches a, then so do* (f +*g)(x*),(fg)(x),(αf)(x), and
(f/g)(x)*(when the limit of g(x) is nonzero).*

*In fact,*

*x→a*lim(f +*g)(x) =* lim

*x→a**f*(x) + lim

*x→a**g(x*),

*x*lim→*a*(αf)(x) = αlim

*x*→*a**f*(x),

*x→a*lim(fg)(x) = lim

*x→a**f*(x)lim

*x→a**g(x),*
*and (when the limit of g(x) is nonzero)*

*x→a*lim

*f*
*g*

(x) = lim*x→a**f*(x)
lim*x*→*a**g(x*).

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem

*Suppose that a*∈ **R, that I is an open interval that contains***a, and that f,g are real functions defined everywhere on I*
*except possibly at a. If f(x) and g(x) converge as x*

*approaches a, then so do* (f +*g)(x*),(fg)(x),(αf)(x), and
(f/g)(x)*(when the limit of g(x) is nonzero).*

*In fact,*

*x→a*lim(f +*g)(x) =* lim

*x→a**f*(x) + lim

*x→a**g(x*),

*x*lim→*a*(αf)(x) = αlim

*x*→*a**f*(x),

*x→a*lim(fg)(x) = lim

*x→a**f*(x)lim

*x→a**g(x),*
*and (when the limit of g(x) is nonzero)*

Theorem (Squeeze Theorem For Functions)
*Suppose that a*∈**R, that I is an open interval that***contains a, and that f,g,h are real functions defined*
*everywhere on I except possibly at a.*

*(i)*

*If g(x*) ≤*h(x*) ≤*f*(x)*for all x* ∈*I*\ {*a*}*, and*

*x→a*lim*f*(x) = lim

*x→a**g(x*) =*L,*
*then the limit of h(x) exists, as x* →*a, and*

*x*lim→*a**h(x*) =*L.*

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Squeeze Theorem For Functions)
*Suppose that a*∈**R, that I is an open interval that***contains a, and that f,g,h are real functions defined*
*everywhere on I except possibly at a.*

*(i)*

*If g(x*) ≤*h(x*) ≤*f*(x)*for all x* ∈*I*\ {*a*}*, and*

*x→a*lim*f*(x) = lim

*x→a**g(x*) =*L,*
*then the limit of h(x) exists, as x* →*a, and*

Theorem (Squeeze Theorem For Functions)
*Suppose that a*∈**R, that I is an open interval that***contains a, and that f,g,h are real functions defined*
*everywhere on I except possibly at a.*

*(i)*

*If g(x*) ≤*h(x*) ≤*f*(x)*for all x* ∈*I*\ {*a*}*, and*

*x→a*lim*f*(x) = lim

*x→a**g(x*) =*L,*
*then the limit of h(x) exists, as x* →*a, and*

*x*lim→*a**h(x*) =*L.*

WEN-CHINGLIEN **Advanced Calculus (I)**

*Suppose that a*∈**R, that I is an open interval that***contains a, and that f,g,h are real functions defined*
*everywhere on I except possibly at a.*

*(i)*

*If g(x*) ≤*h(x*) ≤*f*(x)*for all x* ∈*I*\ {*a*}*, and*

*x→a*lim*f*(x) = lim

*x→a**g(x*) =*L,*
*then the limit of h(x) exists, as x* →*a, and*

Theorem
*(ii)*

*If*|*g(x*)| ≤*M for all x* ∈*I*\ {*a*}*and f*(x) →*0 as x* →*a,*
*then*

*x→a*lim*f*(x)g(x) =0.

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem
*(ii)*

*If*|*g(x*)| ≤*M for all x* ∈*I*\ {*a*}*and f*(x) →*0 as x* →*a,*
*then*

*x→a*lim*f*(x)g(x) =0.

Theorem
*(ii)*

*If*|*g(x*)| ≤*M for all x* ∈*I*\ {*a*}*and f*(x) →*0 as x* →*a,*
*then*

*x→a*lim*f*(x)g(x) =0.

WEN-CHINGLIEN **Advanced Calculus (I)**

Theorem (Comparison Theorem For Functions)
*Suppose that a*∈**R, that I is an open interval that***contains a, and that f,g are real functions defined*
*everywhere on I except possibly at a. If f and g have a*
*limit as x approaches a and*

*f*(x) ≤*g(x),* *x* ∈*I*\ {*a*},
*then*

*x→a*lim*f*(x) ≤ lim

*x→a**g(x*).

Theorem (Comparison Theorem For Functions)
*Suppose that a*∈**R, that I is an open interval that***contains a, and that f,g are real functions defined*
*everywhere on I except possibly at a. If f and g have a*
*limit as x approaches a and*

*f*(x) ≤*g(x),* *x* ∈*I*\ {*a*},
*then*

*x→a*lim*f*(x) ≤ lim

*x→a**g(x*).

WEN-CHINGLIEN **Advanced Calculus (I)**

**Example:**

*For each function f define the positive part of f by*
*f*^{+}(x) = |*f*(x)| +*f*(x)

2 , *x* ∈*Dom(f*),
*and the negative part by*

*f*^{−}(x) = |*f*(x)| −*f*(x)

2 , *x* ∈*Dom(f*).

**Example:**

*For each function f define the positive part of f by*
*f*^{+}(x) = |*f*(x)| +*f*(x)

2 , *x* ∈*Dom(f*),
*and the negative part by*

*f*^{−}(x) = |*f*(x)| −*f*(x)

2 , *x* ∈*Dom(f*).

WEN-CHINGLIEN **Advanced Calculus (I)**

(a)

*Prove that f*^{+}(x) ≥*0, f*^{−}(x) ≥*0, f*(x) =*f*^{+}(x) −*f*^{−}(x), and

|*f*(x)| =*f*^{+}(x) +*f*^{−}(x)*hold for all x* ∈*Dom(f*).(Compare
with Exercise 1,p.11.)

(b)

Prove that if

*L*= lim

*x→a**f*(x)

*exists, then f*^{+}(x) →*L*^{+}*and f*^{−}(x) →*L*^{−} *as x* →*a.*

(a)

*Prove that f*^{+}(x) ≥*0, f*^{−}(x) ≥*0, f*(x) =*f*^{+}(x) −*f*^{−}(x), and

|*f*(x)| =*f*^{+}(x) +*f*^{−}(x)*hold for all x* ∈*Dom(f*).(Compare
with Exercise 1,p.11.)

(b)

Prove that if

*L*= lim

*x→a**f*(x)

*exists, then f*^{+}(x) →*L*^{+}*and f*^{−}(x) →*L*^{−} *as x* →*a.*

WEN-CHINGLIEN **Advanced Calculus (I)**

(a)

*Prove that f*^{+}(x) ≥*0, f*^{−}(x) ≥*0, f*(x) =*f*^{+}(x) −*f*^{−}(x), and

|*f*(x)| =*f*^{+}(x) +*f*^{−}(x)*hold for all x* ∈*Dom(f*).(Compare
with Exercise 1,p.11.)

(b)

Prove that if

*L*= lim

*x→a**f*(x)

*exists, then f*^{+}(x) →*L*^{+}*and f*^{−}(x) →*L*^{−} *as x* →*a.*

(a)

*Prove that f*^{+}(x) ≥*0, f*^{−}(x) ≥*0, f*(x) =*f*^{+}(x) −*f*^{−}(x), and

|*f*(x)| =*f*^{+}(x) +*f*^{−}(x)*hold for all x* ∈*Dom(f*).(Compare
with Exercise 1,p.11.)

(b)

Prove that if

*L*= lim

*x→a**f*(x)

*exists, then f*^{+}(x) →*L*^{+}*and f*^{−}(x) →*L*^{−} *as x* →*a.*

WEN-CHINGLIEN **Advanced Calculus (I)**

**Example:**

Let f,g be real functions, and for each

*x* ∈*Dom(f*) ∩*Dom(g)*define(f∨*g)(x) :=*max{*f*(x),*g(x*)}

and(f ∨*g)(x*) :=min{*f*(x),*g(x*)}.

**Example:**

Let f,g be real functions, and for each

*x* ∈*Dom(f*) ∩*Dom(g)*define(f∨*g)(x) :=*max{*f*(x),*g(x*)}

and(f ∨*g)(x*) :=min{*f*(x),*g(x*)}.

WEN-CHINGLIEN **Advanced Calculus (I)**

(a)

Prove that

(f ∨*g)(x*) = (f +*g)(x*) + |(*f* −*g)(x*)|

2 and

(f ∧*g)(x*) = (f +*g)(x*) − |(*f* −*g)(x*)|

2
*for all x* ∈*Dom(f*) ∩*Dom(g).*

(a)

Prove that

(f ∨*g)(x*) = (f +*g)(x*) + |(*f* −*g)(x*)|

2 and

(f ∧*g)(x*) = (f +*g)(x*) − |(*f* −*g)(x*)|

2
*for all x* ∈*Dom(f*) ∩*Dom(g).*

WEN-CHINGLIEN **Advanced Calculus (I)**

(b)

Prove that if

*L*= lim

*x*→*a**f*(x) *and M* = lim

*x*→*a**g(x*)

exist, then (f ∨*g)(x*) →*L*∨*M and*(f ∧*g)(x*) →*L*∧*M as*
*x* →*a.*

(b)

Prove that if

*L*= lim

*x*→*a**f*(x) *and M* = lim

*x*→*a**g(x*)

exist, then (f ∨*g)(x*) →*L*∨*M and*(f ∧*g)(x*) →*L*∧*M as*
*x* →*a.*

WEN-CHINGLIEN **Advanced Calculus (I)**