Chapter 2: Limits and Derivatives
2.2: The Limit of a Function
Example 1:
a) Graph the function
࠵?
࠵?
=
!
!
!
!
!
!
!
and use
numerical methods
to investigate its behavior near
࠵?
=
2
.
b) Use the
numerical
and
graphical methods
in part (a) to guess the value of
lim
!
→
!
!
!
!
!
!
!
!
.

Intuitive Definition of a Limit:
Suppose
࠵?
(
࠵?
)
is defined when
࠵?
is near the number
࠵?
. (This means that
࠵?
is defined on some open
interval that contains
࠵?
, except possibly at
࠵?
itself.) Then we write
lim
!
→
!
࠵?
࠵?
=
࠵?
and say “the limit of
࠵?
(
࠵?
)
, as
࠵?
approaches
࠵?
, equals
࠵?
”
if we can make the values of
࠵?
(
࠵?
)
arbitrarily close to
࠵?
(as close to
࠵?
as we like) by restricting
࠵?
to be
sufficiently close to
࠵?
(on either side of
࠵?
) but not equal to
࠵?
.
Note:
lim
!
→
!
࠵?
࠵?
=
࠵?
is equivalent to
࠵?
࠵?
→
࠵?
as
࠵?
→
࠵?
, and is read “
࠵?
(
࠵?
)
approaches
࠵?
as
࠵?
approaches
࠵?
”.
Example 2:
Estimate the value of each limit
graphically
.
a)
lim
!
→
!
!"#
!
!
b)
lim
!
→
!
sin
!
!

Definitions: One-Sided Limits
1.
We write
lim
!
→
!
!
࠵?
࠵?
=
࠵?
and say the
left-hand limit of
࠵?
(
࠵?
)
as
࠵?
approaches
࠵?
[or the
limit of
࠵?
(
࠵?
)
as
࠵?
approaches
࠵?
from the left
] is equal to
࠵?
if we can make the values of
࠵?
(
࠵?
)
arbitrarily close to
࠵?
by taking
࠵?
to be sufficiently close to
࠵?
with
࠵?
less than
࠵?
.
Note:
lim
!
→
!
!
࠵?
࠵?
=
࠵?
is equivalent to
࠵?
࠵?
→
࠵?
as
࠵?
→
࠵?
!
.
2.
We write
lim
!
→
!
!
࠵?
࠵?
=
࠵?
and say the
right-hand limit of
࠵?
(
࠵?
)
as
࠵?
approaches
࠵?
[or the
limit of
࠵?
(
࠵?
)
as
࠵?
approaches
࠵?
from the right
] is equal to
࠵?
if we can make the values of
࠵?
(
࠵?
)
arbitrarily close to
࠵?
by taking
࠵?
to be sufficiently close to
࠵?
with
࠵?
greater than
࠵?
.
Note:
lim
!
→
!
!
࠵?
࠵?
=
࠵?
is equivalent to
࠵?
࠵?
→
࠵?
as
࠵?
→
࠵?
!
.
3.
Theorem:
lim
!
→
!
࠵?
࠵?
=
࠵?
if and only if
lim
!
→
!
!
࠵?
࠵?
=
࠵?
and
lim
!
→
!
!
࠵?
࠵?
=
࠵?
.
Example 3:
Use the graph below to evaluate each limit, or if appropriate indicate that the limit does not exist.
a)
lim
!
→
!
!
!
࠵?
࠵?
=
b)
lim
!
→
!
!
!
࠵?
࠵?
=
c)
lim
!
→
!
!
࠵?
࠵?
=
d)
lim
!
→
!
!
!
࠵?
࠵?
=
e)
lim
!
→
!
!
!
࠵?
࠵?
=
f)
lim
!
→
!
!
࠵?
࠵?
=
g)
lim
!
→
!
!
࠵?
࠵?
=
h)
lim
!
→
!
࠵?
࠵?
=
i)
lim
!
→
!
!
࠵?
࠵?
=
j)
lim
!
→
!
!
࠵?
࠵?
=
k)
lim
!
→
!
࠵?
࠵?
=
l)
lim
!
→
!
࠵?
࠵?
=

Definitions: Intuitive Definition of Infinite Limits
Definition: Vertical Asymptote
The vertical line
࠵?
=
࠵?
is called a
vertical asymptote
of the curve
࠵?
=
࠵?
(
࠵?
)
if at least one of the following
statements is true:
lim
!
→
!
࠵?
࠵?
=
∞
lim
!
→
!
!
࠵?
࠵?
=
∞
lim
!
→
!
!
࠵?
࠵?
=
∞
lim
!
→
!
࠵?
࠵?
=
−
∞
lim
!
→
!
!
࠵?
࠵?
=
−
∞
lim
!
→
!
!
࠵?
࠵?
=
−
∞
1.
Let
࠵?
be a function defined on both sides of
࠵?
, except
possibly at
࠵?
itself. Then
lim
!
→
!
࠵?
(
࠵?
)
=
∞
means that the values of
࠵?
(
࠵?
)
can
be made arbitrarily large (as large as we please) by
taking
࠵?
sufficiently close to
࠵?
, but not equal to
࠵?
.
Alternatively,
lim
!
→
!
࠵?
(
࠵?
)
=
∞
can be expressed in
the following manner:
࠵?
(
࠵?
)
→
∞
as
࠵?
→
࠵?
.
Note:
This does not mean that we are regarding
as a
number. Nor does it mean that the limit exists. It
simply expresses the particular way in which the limit
does not exist:
࠵?
(
࠵?
)
grows without bound as
࠵?
approaches
࠵?
.
2.
Let
࠵?
be a function defined on both sides of
࠵?
, except
possibly at
࠵?
itself. Then
lim
!
→
!
࠵?
(
࠵?
)
=
−
∞
means that the values of
࠵?
(
࠵?
)
can be made arbitrarily large negative by
taking
࠵?
sufficiently close to
࠵?
, but not equal to
࠵?
.