Ans the Following Question plzzzzz....... Evaluate the limit along the paths                                (a) the y-axis (x = 0),                             (b) the x-axis ( y = 0 ),                              (c) the line y = mx for all value of .   Lim x,y -->(0,0) [x^2-y^3/ x^2+y^2]

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Let f be `f(x,y)=(x^2-y^3)/(x^2+y^2)`

a) Along the y axis, x=0

`f(0,y)=(0-y^3)/ (0+y^2)=-y`

 

`lim_(y rarr 0, x=0)f(x,y)=lim_(y rarr0)-y=0`

First limit `lim_(y rarr 0, x=0)f(x,y)=0`

b) Along the x axis, y=0

`f(x,0)=x^2/x^2=1`

`lim_(x rarr 0, y=0) f(x,y)=lim_(x rarr 0, y=0)1=1`

Second limit `lim_(x rarr 0, y=0) f(x,y)=1`

C) Along a line y=mx

`f(x,y)=f(x,mx)=(x^2-m^3x^3)/(x^2+m^2x^2)=((1-m^3x)x^2)/((1+m^2)x^2)`

`f(x,y)=(1-m^3x)/(1+m^2)`

`lim_(x rarr 0, y=mx) f(x,y)=lim_(x rarr 0, y=mx)(1-m^3x)/(1+m^2)=1/(1+m^2).`

Third limit `lim_(x rarr 0, y=mx) f(x,y)=1/(1+m^2).`

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