A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.
Limits are written in the following form:
The expression above states that when approaches arbitrarily close to , the function becomes arbitrarily close to the value , which is called the limit.
We can examine the limit of a simple continuous function, .
The limit at x=0 of f(x) = x^2 would pretty clearly be 0, since .
Indeed for this function, . But this is a special case, in the majority of limits cannot be solved in this manner.
For a very different example; given
(as pictured below)
The limit of because x approaches 0 is 0 (just as in ), but ; is not continuous at (as shown on the right).
In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.
Here we look at one such case:
(as pictured above)
Rigorous Definition of Limit
This definition is more appropriate for 2nd year calculus students and higher.
Properties of Limits