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 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 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.