A joint limit does not mean the iterated limits exist. Consider xn,m:=min{n,m}(−1)n+m.
a) Show that for no n does limm→∞xn,m exist, and for no m does limn→∞xn,m exist. So neither limn→∞limm→∞xn,m nor limm→∞limn→∞xn,m makes any sense at all.
b) Show that the joint limit of {xn,m}n,m=1∞ exists and equals 0.