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}


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


[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 onesided 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.