Let #f(x)# be a real function of real variable defined in the domain #I sub RR#.
Let now #barx in RR# be a generic real number.
If #bar x in I# then the function has a value in that point: #f(bar x)#.
On the other hand, the limit:
#lim_(x->barx) f(x)#
is defined if #x # is a point of accumulation for #I#, which means that in every interval #(a,b)# such that #barx in (a,b)# we can find a point #xi in I nn (a,b)#. This is possible also if #barx notin I#
For example consider the function:
#f(x) = ln(1+x)/x#
Here #I = (-1,0) uu (0,+oo)#, so the function is not defined for #x=0#. Hower #x=0# is a point of accumulation for #I# because in any interval #(-delta, delta)# we can find points where #f(x)# is defined. (actually all of them except #x=0#).
In fact:
#lim_(x->0) ln(1+x)/x = 1#
A more extreme example is to consider a function #phi(q)# defined only for rational numbers, #q in QQ#.
We could still evaluate #lim_(q->sqrt2) phi(q)#, because though #sqrt2# is irrational, it can be approximated with a rational number as closely as we want.
By the way the converse is also true. The limit:
#lim_(x->barx) f(x)#
cannot be defined even if #bar x in I#, if #barx# is not a point of accumulation for #I#.
