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 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 = Cn |
|
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
x1 which is very close to
x = a, there always exists
another
x-value
x0 closer to
a such that
f (x0) is closer to
f (a)
than
f (x1).
A limit of a function can also be taken "from the left" and "from the right."
These are called one-sided limits. The equation
xâƒa-]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
xâƒa+]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
xâƒa-]f (x)≠xâƒa+]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.
= 
if a≥ 0
= 
, if D≠ 0
