What is the limit of (x^3-y^3)/(x^2+y^2) as (x,y)-->(0,0)

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We have to find the value of lim x,y-->0 [(x^3-y^3)/(x^2+y^2)]

We cannot substitute x and y with 0 in the expression as that would yield 0/0 which is indeterminate.

We see that the expression is defined for all values of (x,y) except (0,0), so we approach (0,0) along y = mx

lim x,y-->0 [(x^3-y^3)/(x^2+y^2)]

=> lim x-->0 [(x^3- m^3*x^3)/(x^2+m^2*x^2)]

=> lim x-->0 [(x^3(1 - m^3))/(x^2(1+m^2))]

=> lim x-->0 [(x(1 - m^3))/(1+m^2)]

Now substituting x = 0

=> 0*(1 - m^3) / (1 + m^2)

=> 0

Therefore the required value of the limit is 0.

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