Limits of a Function

In mathematics, limits of a function is a fundamental concept in calculus, as it is used to define continuity, derivatives and integrals.

A function has a limit of L if f(x) approaches L when x approaches a certain value.
In general, a function f assigns a value f(x) to a variable x. Suppose we are given a range of values of x such as the domain 1 <= x <= 0.01 for the function f(x) = (sin x)/x. The following shows the table of values for the function. The function f(x) = (sin x)/x approaches 2 when x approaches 0.01. By definition, the function f(x) = (sin x)/x has a limit of 2. Let the value of x=0.01 be p, and the value of f(x) when x=0.01 be L, the expression that defines limit is:



Definition of Continuity Using Limits
A variable x may approach a certain value p from above(right) or below(left). The expression which defines each of these cases are: If both of this limits are unequal, then we have a discontinuity between two graphs i.e. these two graphs does not intersect a common line. We have the definition of continuity as:
 * Above
 * Below

Continuity is the property of two graphs that these graphs intersect a common line x = k, where k is any suitable value of x.
If two graphs have discontinuity, their limits does not exist i.e. the expression that f(x) approaches L when x approaches p by above or below are unequal.

The diagram on the left shows the discontinuity of two graphs. Both graphs does not intersect the line x = x0. Both the limits where x approaches x0 from above and below are unequal.