MATH0201 BASIC CALCULUS
MATH0201
BASIC CALCULUS
Limits
Dr. WONG Chi Wing
Department of Mathematics, HKU
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Evaluation of Limits
Indeterminate Form
Continuity
Continuous Functions
Intermediate Value Theorems
Infinite Limits and Limits at Infinity
Infinite Limits
Limits at Infinity
Reference
§ 1.5–6 of the textbook.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Definition 1 (Limit at a Point (p.64))
Let
f
be a function and
c
∈
R
. We write
lim
x
→
0
f
(
x
) =
L
or
f
(
x
)
→
L
as
x
→
c
and call
L
the
limit of f at c
if
f
(
x
)
gets closer and closer to
L
as
x
gets closer and closer to
c
from both sides,
but not equal to
c
.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Example 2 (Graphical Approach)
Consider the graph of
f
(
x
) =
x
,
x
6
=
0.
We have lim
x
→
0
f
(
x
) =
0.
lim
x
→
0

x
=
lim
x
→
0
+
f
(
x
) =
0
.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Example 3 (Graphical Approach)
Consider the graph of
f
(
x
) =

x

/
x
,
x
6
=
0.
lim
x
→
0
f
(
x
)
DOES NOT exist.
lim
x
→
0


x

x
=

1
and
lim
x
→
0
+

x

x
=
1
.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Definition 4 (One–sided Limit at a Point: Left Hand Limit
(p.79))
Let
f
be a function and
c
∈
R
. We write
lim
x
→
c

f
(
x
) =
L
and call
L
the
left hand limit of f at c
if
f
(
x
)
gets closer and
closer to
L
as
x
gets closer and closer to
c
from the left hand
side,
but not equal to
c
.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Definition 5 (One–sided Limit at a Point: Right Hand Limit
(p.79))
Let
f
be a function and
c
∈
R
. We write
lim
x
→
c
+
f
(
x
) =
L
and call
L
the
right hand limit of f at c
if
f
(
x
)
gets closer and
closer to
L
as
x
gets closer and closer to
c
from the right hand
side,
but not equal to
c
.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Remark 1
lim
x
→
c
f
(
x
)
may also be called an
two–sided limits
.
Theorem 6 (Existence of a Limit (p.80))
The (two–sided) limit
lim
x
→
c
f
(
x
)
exists and equals L if and
only if both the one–sided limits
lim
x
→
c

f
(
x
)
and
lim
x
→
c
+
f
(
x
)
exist and they are equal to L.
MATH0201 BASIC CALCULUS
Limits at a Point
Two–sided and One–sided Limits
Example 7
Below is the graph a function
g
(
x
)
.
MATH0201 BASIC CALCULUS
Limits at a Point
Evaluation of Limits
Theorem 8 (Algebraic Properties of Limits (p.66))
Let f and g be functions, and assume that
lim
x
→
c
f
(
x
) =
L
and
lim
x
→
c
g
(
x
) =
M
.
1.
For any real k,
lim
x
→
c
k
=
k; and
lim
x
→
c
x
=
c.
2.
lim
x
→
c
[
f
(
x
)
±
g
(
x
)] =
L
±
M.
3.
For any real number k,
lim
x
→
c
kf
(
x
) =
kL.
4.
lim
x
→
c
f
(
x
)
g
(
x
) =
LM.
5.
If M
6
L
M
.