Is $f(x,y) = \frac{x^ny^m}{x^{n'}+y^{m'}}$ continuous at origin?
Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$