Section 1.3: The Limit of a Function
Brett Geiger
Southern Methodist University
Math 1337 - Calculus I
Brett Geiger
Section 1.3: The Limit of a Function

Limit Intuition
Consider the function below:
Brett Geiger
Section 1.3: The Limit of a Function

Limit Intuition
Consider the function below:
Technically speaking, the function has a “hole” at
x
=
a
, so we know the
function is not technically defined at
x
=
a
.
Brett Geiger
Section 1.3: The Limit of a Function

Limit Intuition
Consider the function below:
Technically speaking, the function has a “hole” at
x
=
a
, so we know the
function is not technically defined at
x
=
a
. Realistically though, this
feels a bit silly because it clearly looks like the function value “near
x
=
a
” is about
L
, whatever that value is.
Brett Geiger
Section 1.3: The Limit of a Function

Limit Intuition
Consider the function below:
Technically speaking, the function has a “hole” at
x
=
a
, so we know the
function is not technically defined at
x
=
a
. Realistically though, this
feels a bit silly because it clearly looks like the function value “near
x
=
a
” is about
L
, whatever that value is. Limits help to make this line
of thinking rigorous.
Brett Geiger
Section 1.3: The Limit of a Function

Limit Intuition
Consider the function below:
Technically speaking, the function has a “hole” at
x
=
a
, so we know the
function is not technically defined at
x
=
a
. Realistically though, this
feels a bit silly because it clearly looks like the function value “near
x
=
a
” is about
L
, whatever that value is. Limits help to make this line
of thinking rigorous. The concept of limits is the entire backbone of
Calculus, and surprisingly (at least in my opinion), is the most important
concept going forward as it is the basis of essentially all definitions.
Brett Geiger
Section 1.3: The Limit of a Function

Definition of a Limit
Definition:
Suppose
f
(
x
) is defined when
x
is near the number
a
(but
not necessarily at
x
=
a
.
).
Brett Geiger
Section 1.3: The Limit of a Function

Definition of a Limit
Definition:
Suppose
f
(
x
) is defined when
x
is near the number
a
(but
not necessarily at
x
=
a
.
). Then, we write
lim
x
→
a
f
(
x
) =
L
and say
the limit of
f
(
x
)
,
as
x
approaches
a
,
is
L
if the function
values
f
(
x
) become arbitrarily close to
L
as we take
x
-values arbitrarily
close to
a
on both sides (but not equal to
a
).
Brett Geiger
Section 1.3: The Limit of a Function

Definition of a Limit
Definition:
Suppose
f
(
x
) is defined when
x
is near the number
a
(but
not necessarily at
x
=
a
.
). Then, we write
lim
x
→
a
f
(
x
) =
L
and say
the limit of
f
(
x
)
,
as
x
approaches
a
,
is
L
if the function
values
f
(
x
) become arbitrarily close to
L
as we take
x
-values arbitrarily
close to
a
on both sides (but not equal to
a
).
Notes:
(a) Notice that the definition does not require
f
(
x
) to even exist at
x
=
a
, only for values near
a
.

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