how to show limit exist or not

[junhoma]

question,

Show that limit (x,y) approaches (0,0) x^2+y^3/x^2+y does not exist

I tried in this way
lets approach (0,0) along x axis
implies that x will vary and y =0

so when x approaches 0 x^2+y^3/x^2+y=x^2/x^2=1

when y approaches 0 x=0
So x^2+y^3/x^2+y= y^3/y=y^2 if we put y= 0 then = 0

both xapproaching 0 =1 and y approaching 0= 0 are not equal so limit doesnot exist.
Am i right please help me in detail

also can you please at what condition I can use like x=y, y=x^2 and y= mx etc to show from different angle

Please help me

 

[galactus]

Please use grouping symbols (parnetheses).

Is it this:

\(\displaystyle \lim_{(x,y)\ to (0,0)}\frac{x^{2}+y^{3}}{x^{2}+y}\)?.

If so, it consists of all points in the xy-plane except for (0,0).

Think about approaching (0,0) along two different paths.

Along the x-axis, every point is of the form (x,0) and the limit is

\(\displaystyle \lim_{(x,0)\to (0,0)}\left(\frac{x^{2}+0}{x^{2}+0}\right)=\lim_{(x,0)\to (0,0)}(1)=1\)

Now, if it approaches (0,0) along the line y=x, then:

\(\displaystyle \lim_{(x,x) \to (0,0)}\left(\frac{x^{2}+x^{3}}{x^{2}+x}\right)=\lim_{(x,x) \to (0,0)}x=0\)

Two different results, thus it does not exist.