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.