# help with limit and lines passing through the origin

I need to check that there exists a limit at (0,0) for any line passing through the origin but the limit does not exist at the origin.

$f(x,y)={y \over y + x^2}$ if $y \neq -x^2$ otherwise $f(x, -x^2) = 1$

My question is regarding how I can make sure that I'm checking for all the lines passing through the origin.

At the moment I'm doing the following. We approach the origin through $y=ax$

$\lim_{(x,y=ax) \to (0,0)} f(x,y)={y \over y + x^2 }=\lim_{(x,y=ax)}{ax \over ax + x^2 }$

As $\lim_{(x,y=ax) \to (0,0)} ax=0$ and $\lim_{(x,y=ax) \to (0,0)} ax +x^2=0$ I can use L'Hopital and we get:

$\lim_{(x,y=ax) \to (0,0)} f(x,y)={y \over y + x^2 }=\lim_{(x,y=ax)\to (0,0)}{ax \over ax + x^2 }=\lim_{(x,y=ax)\to (0,0)}{a \over a + 2x }=1$

I can also approach $x=ay$. But my question is whether I'm checking for all the lines that go though the origin by doing this. I'm under the impression that I'm missing more lines.

Edit: To put my question other way. Given the above function f(x,y), can I get a formula of all the straight lines going through the origin. I'm not sure what I'm saying makes any sense so I apologize in advance.

• You can guess a limit and try to prove it using $\epsilon-\delta$ definition.Then you can be sure that you cover all the lines(if the limit exists). – Abhra Abir Kundu Jun 4 '13 at 18:02
• @AbhraAbirKundu line chhara onno curve borabor gele cholbena? – Marso Jun 4 '13 at 18:04
• If the limit exists then it must exist along all the curves. So you can do it using some other curve also. – Abhra Abir Kundu Jun 4 '13 at 18:05
• When you assume $y=ax$, $a\neq 0$, you are checking all the lines passing through origin except the $x$-axis and $y$-axis. – Shuhao Cao Jun 6 '13 at 5:58

Hint: what about $y=mx^2$? you will get different limit for different $m$
• But $y=mx^2$ is not a line,it is parabola, right? – jjjx Jun 4 '13 at 18:04