Continuity and Limits
Defining a Limit
A limit of a function is the value that function approaches as the
independent variable of the function approaches a given value. The equation
f (x) = t
is equivalent to the statement "The limit of
f
as
x
goes to
c
is
t
." Another way to phrase this equation is "As
x
approaches
c
, the value of
f
gets arbitrarily close to
t
." This is the
essential concept of a limit.
Here are some properties of limits.
x = a
|
|
k = k, where k is a constant.
|
|
x
b = a
b
|
 =  if a≥ 0
|
Here are some properties of operations with limits. Let
f (x) = C
, and
g(x) = D
.
|
kf (x) = kC, where k is a constant.
|
|
(f (x)±g(x)) = C±D
|
|
f (x)×g(x) = C×D
|
 =  , if D≠ 0
|
|
[f (x)]n = C
n
|
The more formal definition of a limit is the following.
f (x) = A
if and only if for any positive number
ε
, there exists another
positive number
δ
, such that if
0 < | x - a| < ε
, then
| f (x) - A| < δ
. This definition basically states that if
A
is the limit of
f
as
x
approaches
a
, then any time
f (x)
is within
ε
units of a
value
A
, another interval
(x - δ, x + δ)
exists such that all
values of
f (x)
between
(x - δ)
and
(x + δ)
lie within the bounds
(A - ε, A + ε)
. A simpler way of saying it is this: if you
choose an
x
-value
x
1
which is very close to
x = a
, there always exists
another
x
-value
x
0
closer to
a
such that
f (x
0)
is closer to
f (a)
than
f (x
1)
.
A limit of a function can also be taken "from the left" and "from the right."
These are called one-sided limits. The equation
xâÜ’a
-]f (x) = A
reads "The limit of
f (x)
as
x
approaches
a
from the left is
A
."
"From the left" means from values less than
a
-- left refers to the left side
of the graph of
f
. The equation
xâÜ’a
+]f (x) = A
means that the
limit is found by calculating values of
x
that approach
a
which are greater
than
a
, or to the right of
a
in the graph of
f
.
There are a few cases in which a limit of a function
f
at a given
x
-value
a
does not exist. They are as follows: 1) If
xâÜ’a
-]f (x)≠
xâÜ’a
+]f (x)
. 2) If
f (x)
increases of decreases without bound as
x
approaches
a
. 3) If
f
oscillates (switches back and forth) between
fixed values as
x
approaches
a
. In these situations, the limit of
f (x)
at
x = a
does not exist.
One of the most important things to remember about limits is this:
f (x)
is independent of
f (a)
. All that matters is the behavior of the
function at the
x
-values near
a
, not at
a
. It is not uncommon for
a function have a limit at an
x
-value for which the function is undefined.
