# Ans the Following Question plzzzzz.......Evaluate the limit along the paths (a) the y-axis (x = 0), ...

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]

### 1 Answer | Add Yours

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).`