For the limit lim_((x,y) -> (1,0))(xy-y)/((x-1)^(2)+y^(2)) We will replace y with m(x-1), to get lim_(x -> 1)(x*m(x-1)-m(x-1))/((x-1)^(2)+m^(2)(x-1)^(2)) We can cancel (x-1) from the numerator and the denominator =lim_(x -> 1)(x*m-m)/((x-1)+m^(2)(x-1)) =lim_(x -> 1)(m(x-1))/((x-1)+m^(2)(x-1)) We can again cancel (x-1) from the numerator and the denominator -lim_(x -> 1)(m)/(1+m^(2))=(m)/(1+m^(2)) Since the limit is not independent of m, it does not exist Result: The limit does not exist