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.

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

Hint: what about $y=mx^2$? you will get different limit for different $m$

  • $\begingroup$ But $y=mx^2$ is not a line,it is parabola, right? $\endgroup$ – jjjx Jun 4 '13 at 18:04
  • $\begingroup$ you need to show limit doesn't exists right? $\endgroup$ – Marso Jun 4 '13 at 18:05
  • $\begingroup$ Yes, and I know I can do that by showing that we get different limits with a parabola or whatever. What I would like to know is whether I can make sure that I'm checking all the straight lines. $\endgroup$ – jjjx Jun 4 '13 at 18:08
  • $\begingroup$ okay I am withdrawing my answer! $\endgroup$ – Marso Jun 4 '13 at 18:09

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