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.||
|xb = ab||
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 xâÜa-]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 xâÜa+]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 xâÜa-]f (x)≠xâÜa+]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 neara, not at a. It is not uncommon for
a function have a limit at an x-value for which the function is undefined.