Find the limit, if it exists, or show that the limit does not exist. lim (x,y) tends to (1,0) xy-y/(x-1)^(2)+y^(2)

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Best Answer

**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