Limit
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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.
Contents 
Informal Definition
We can consider the idea of limits using a simple continuous function, .
We want to examine the limit of x= 0 for this function. Since this graph is a simple unbroken line, we realize that
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
Ideas for the Future
 an interactive diagram in which changing the size of epsilon shows a corresponding delta, or something.
For More Information
More examples  http://archives.math.utk.edu/visual.calculus/1/definition.6/index.html
http://mathworld.wolfram.com/Limit.html
Even more mathematical description 