If f(x) = , For what integers m and n does both and exist?

For a limit of exist at a point, left hand limit most be equal to the right hard limit at that point.

Now, for to exist, we have

=

         = m. 0+n = n

=

         = n.0+m = m

Hence, n = m. Thus exist for all m = n.

For to exist, we have

=

         = n(1 + 0) + m = m + n

= f(l +n)

         = n(1 + 0) + m = m + n

As m + m = m + n,

exists for all integral values of m and n.