Problem :
Define a function by f (x) = x  3 for x≠1 and let f (1) = 1. At which values of x
is f (x) continuous? Does f (x) have a limit at x = 1? If so, what is this limit?
A graph of this function is displayed below.
Figure %: Plot of f (x) = x  3 for x≠1 and let f (1) = 1
As
x approaches
1, the values of
f (x) approach
1  3 =  2, so
f (x) =  2. 

However,
f (1) = 1≠  2. Therefore,
f (x) is not continuous at
x = 1. It is clear from
the graph that
f (x) is continuous at all other values of
x.
Problem :
Consider the function
f (x) = , 

defined for
x≠  1, 0. Does
f (x) have a limit at
x = 0? If so, what is the limit?
Multiply the numerator and denominator of the expression defining
f (x) by
x^{2} to obtain
g(x) = 

which will be equal to
f (x) for all
x≠ 0. This new function clearly has limit
2 at
x = 0, thus so does
f (x). It does not make sense to ask whether
f (x) is continuous at
0, because
0 is not in its domain.