The typical mathematical definition of a limit for functions is the #epsilon-delta# definition, and is as follows:
Given a subset #S# of #RR#, numbers #c in S#, #L in RR#, and a function #f:S->RR#, we say that "the limit* as #x# approaches #c# of #f(x)# is #L# if for every #epsilon > 0# there exists #delta > 0# such that #0<|x-c|< delta# implies #|f(x)-f(c)| < epsilon#
We denote this #lim_(x->c)f(x) = L#
*#c# does not actually have to be in #S# so long as there are values in #S# which are arbitrarily close to #c#. For example, if #S=(0,1)# we could still use #0# or #1# as #c#.
Intuitively, this is saying that we can make #f(x)# arbitrarily close to #f(c)# by making #x# close (but not necessarily equal) to #c#.
There are also versions which allow for #c = +-oo#. To handle #c->oo# we change the portion saying that "there exists #delta>0# such that if #0<|x-c|< delta|#..." to saying that "there exists #N>0# such that if #x>N# then #|f(x)-f(c)| < epsilon#." The #-oo# case is similar.
There are other types of limits out there as well. For example, we can consider the limit of a sequence #(a_n)# as #n->oo#. The definition is similar to the #oo# variation above, where we, in more formal language, basically say that the limit is #L# if we an make #a_n# be close to #L# by making #n# grow large.
The concept of limits is extremely important to calculus, and is worth spending the time to understand. In addition to knowing the definition, it is useful to have an intuition as to what is actually going on. This can be gained by working on examples, but also looking into teaching materials available in textbooks and online.
