# Limits and Continuity for Vector Valued Functions

```Limits
Continuity
Limits and Continuity for Vector Valued
Functions
Bernd Schröder
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Introduction
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Introduction
1. Although we mostly use derivatives and integrals in
calculus
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Introduction
1. Although we mostly use derivatives and integrals in
calculus, all concepts in calculus can formally be traced
back to limits.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Introduction
1. Although we mostly use derivatives and integrals in
calculus, all concepts in calculus can formally be traced
back to limits.
2. So we must start the calculus of vector-valued functions
with limits.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Introduction
1. Although we mostly use derivatives and integrals in
calculus, all concepts in calculus can formally be traced
back to limits.
2. So we must start the calculus of vector-valued functions
with limits.
3. However, limits are cumbersome.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Introduction
1. Although we mostly use derivatives and integrals in
calculus, all concepts in calculus can formally be traced
back to limits.
2. So we must start the calculus of vector-valued functions
with limits.
3. However, limits are cumbersome.
4. Thankfully, a lot of calculus of vector valued functions can
be reduced to doing calculus with the components.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.)
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.) Let p be
a real number.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.) Let p be
a real number. The function f is said to converge to L at p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.) Let p be
a real number. The function f is said to converge to L at p if
and only if, for all x sufficiently close to but not equal to p,
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.) Let p be
a real number. The function f is said to converge to L at p if
and only if, for all x sufficiently close to but not equal to p, we
have that f (x) is close to L.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.) Let p be
a real number. The function f is said to converge to L at p if
and only if, for all x sufficiently close to but not equal to p, we
have that f (x) is close to L.
L will be called the limit of the function at p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
All Limits are Defined the Same Way
When the input gets close to the place where we take the limit,
the output gets close to the limit itself. Recall:
Definition. (Informal definition of the limit at a point.) Let p be
a real number. The function f is said to converge to L at p if
and only if, for all x sufficiently close to but not equal to p, we
have that f (x) is close to L.
L will be called the limit of the function at p, denoted
L = lim f (x).
x→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.)
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
u
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
u
L
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
u
L
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
7
u
L
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
ε
7
u
L
lim~c(t) = ~L.
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
lim~c(t) = ~L.
ε
7
u
]
JJ
L
J
J
J
J
J
J
J
J
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
lim~c(t) = ~L.
ε
7
u
]
JJ
L
J
J
J
J
J
J
J J
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
lim~c(t) = ~L.
ε
7
u
]
JJ
L
J
J
J
J
~c(p − δ ) J
J
J J
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is
the limit of the vector valued
function ~c(t) at p if and only
if, for all t that are sufficiently
close to but not equal to p, we
have that~c(t) is close to ~L. In
symbols, we write
lim~c(t) = ~L.
ε
7
u
]
JJ
L
J
J
J
J
~c(p − δ ) J
~c(p + δ )
J
J J
t→p
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=

.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=
,
.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
where the equation is to be read as “the limit on the left side
exists if and only if all limits on the right side exist.”
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=
,
.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
where the equation is to be read as “the limit on the left side
exists if and only if all limits on the right side exist.”
Explanation.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=
,
.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
where the equation is to be read as “the limit on the left side
exists if and only if all limits on the right side exist.”
Explanation.
~c(t) −~L
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=
,
.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
where the equation is to be read as “the limit on the left side
exists if and only if all limits on the right side exist.”
Explanation.
~c(t) −~L

 
c1 (t)
L1
 c2 (t)   L2
 

= 
..  −  ..

.   .
cn (t)
Ln
Bernd Schröder
Limits and Continuity for Vector Valued Functions





Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=
,
.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
where the equation is to be read as “the limit on the left side
exists if and only if all limits on the right side exist.”
Explanation.
~c(t) −~L

 

c1 (t)
L1  c2 (t)   L2 
 


= 
..  −  .. 

.   . 
cn (t)
Ln q
2
2
2
=
c1 (t) − L1 + c2 (t) − L2 + &middot; &middot; &middot; + cn (t) − Ln
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Proposition. The limit of a vector valued function~c(t) can be
taken componentwise. That is,

 

c1 (t)
limt→p c1 (t)
 c2 (t)   limt→p c2 (t) 

 

lim~c(t) = lim 
=
,
.
.
.
.
t→p
t→p 
.  
. 
cn (t)
limt→p cn (t)
where the equation is to be read as “the limit on the left side
exists if and only if all limits on the right side exist.”
Explanation.
~c(t) −~L

 

c1 (t)
L1  c2 (t)   L2 
 


= 
..  −  .. 

.   . 
cn (t)
Ln q
2
2
2
=
c1 (t) − L1 + c2 (t) − L2 + &middot; &middot; &middot; + cn (t) − Ln
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Example.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

3 − 2t2

Example. Compute the limit lim 
t2 −t
t
cos(t)−1
t
t→0


 or show that it
does not exist.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

3 − 2t2

Example. Compute the limit lim 
t2 −t
t
cos(t)−1
t
t→0


 or show that it
does not exist.


lim 
t→0
3 − 2t2
t2 −t
t
cos(t)−1
t
Bernd Schröder
Limits and Continuity for Vector Valued Functions



Louisiana Tech University, College of Engineering and Science
Limits
Continuity

3 − 2t2

Example. Compute the limit lim 
t2 −t
t
cos(t)−1
t
t→0


 or show that it
does not exist.


lim 
t→0
3 − 2t2
t2 −t
t
cos(t)−1
t
Bernd Schröder
Limits and Continuity for Vector Valued Functions


3 − 2t2

t−1 
 = lim 
t→0
− sin(t)

Louisiana Tech University, College of Engineering and Science
Limits
Continuity

3 − 2t2

Example. Compute the limit lim 
t2 −t
t
cos(t)−1
t
t→0


 or show that it
does not exist.


lim 
t→0
3 − 2t2
t2 −t
t
cos(t)−1
t
Bernd Schröder
Limits and Continuity for Vector Valued Functions


3 − 2t2

t−1 
 = lim 
t→0
− sin(t)


3
= lim  −1 
t→0
0

Louisiana Tech University, College of Engineering and Science
Limits
Continuity

3 − 2t2

Example. Compute the limit lim 
t2 −t
t
cos(t)−1
t
t→0


 or show that it
does not exist.


lim 
t→0
3 − 2t2
t2 −t
t
cos(t)−1
t
Bernd Schröder
Limits and Continuity for Vector Valued Functions


3 − 2t2

t−1 
 = lim 
t→0
− sin(t)


3
= lim  −1 
t→0
0

Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function. Then f is called continuous at
a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function. Then f is called continuous at
a if and only if the following three conditions are satisfied.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function. Then f is called continuous at
a if and only if the following three conditions are satisfied.
1. f (a) is defined.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function. Then f is called continuous at
a if and only if the following three conditions are satisfied.
1. f (a) is defined.
2. lim f (x) exists.
x→a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function. Then f is called continuous at
a if and only if the following three conditions are satisfied.
1. f (a) is defined.
2. lim f (x) exists.
x→a
3. lim f (x) = f (a).
x→a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Continuity is Always Defined the Same Way
The limit at the point must be equal to the value at the point.
Recall:
Definition. Let f be a function. Then f is called continuous at
a if and only if the following three conditions are satisfied.
1. f (a) is defined.
2. lim f (x) exists.
x→a
3. lim f (x) = f (a).
x→a
If f is continuous at every point a in the set A, then f is called
continuous on A.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Proposition.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Proposition. The vector valued function~c(t) is continuous at a
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Proposition. The vector valued function~c(t) is continuous at a
if and only if all its components are continuous at a.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Proposition. The vector valued function~c(t) is continuous at a
if and only if all its components are continuous at a. Moreover,
a vector valued function is continuous on the set S
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Proposition. The vector valued function~c(t) is continuous at a
if and only if all its components are continuous at a. Moreover,
a vector valued function is continuous on the set S if and only if
all its components are continuous on the set S.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Definition. The vector valued function~c(t) is called
continuous at a if and only if the following three conditions are
satisfied.
1. ~c(a) is defined,
2. lim~c(t) exists,
t→a
3. lim~c(t) =~c(a).
t→a
The vector valued function~c(t) will be called continuous on
the set S if and only if~c(t) is continuous at every point in the set
S.
Proposition. The vector valued function~c(t) is continuous at a
if and only if all its components are continuous at a. Moreover,
a vector valued function is continuous on the set S if and only if
all its components are continuous on the set S.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity
Example.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component is continuous on its domain.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component
is continuous on its domain.
I ln 9 − t2 is continuous on (−3, 3).
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component
is continuous on its domain.
I ln 9 − t2 is continuous on (−3, 3).
I 1 is continuous on (−∞, 2) ∪ (2, ∞)
2−t
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component
is continuous on its domain.
I ln 9 − t2 is continuous on (−3, 3).
I 1 is continuous on (−∞, 2) ∪ (2, ∞)
2−t
√
I
t + 1 is continuous on [−1, ∞).
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component
is continuous on its domain.
I ln 9 − t2 is continuous on (−3, 3).
I 1 is continuous on (−∞, 2) ∪ (2, ∞)
2−t
√
I
t + 1 is continuous on [−1, ∞).
The function~c(t) is continuous on the intersection of these
domains.
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component
is continuous on its domain.
I ln 9 − t2 is continuous on (−3, 3).
I 1 is continuous on (−∞, 2) ∪ (2, ∞)
2−t
√
I
t + 1 is continuous on [−1, ∞).
The function~c(t) is continuous on the intersection of these
domains. So it is continuous on [−1, 2) ∪ (2, 3).
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
Limits
Continuity

Example. Determine where the function~c(t) = 
ln 9 − t2
1

√ 2−t
t+1

is continuous.
Every component
is continuous on its domain.
I ln 9 − t2 is continuous on (−3, 3).
I 1 is continuous on (−∞, 2) ∪ (2, ∞)
2−t
√
I
t + 1 is continuous on [−1, ∞).
The function~c(t) is continuous on the intersection of these
domains. So it is continuous on [−1, 2) ∪ (2, 3).
Bernd Schröder
Limits and Continuity for Vector Valued Functions
Louisiana Tech University, College of Engineering and Science
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