Timeline for What's the best way to explain multivariable limit problems to students who are not familiar with $\epsilon$ - $\delta$ proofs?

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May 23, 2017 at 3:27	comment	added	Steven Gubkin		Sure. But there are some simple examples which can convey the problem. For instance, $f(x,y)=0$ if $y=x^2$ with $x \neq 0$, and $f(x,y) = 1$ otherwise. This "clearly" has one limit along each line, and a different limit along the parabola.
May 23, 2017 at 3:23	comment	added	mweiss		@StevenGubkin However, if one is trying to do this without the epsilon-delta definition, one's hands are somewhat tied.
May 23, 2017 at 3:23	comment	added	mweiss		@StevenGubkin Yes, that's certainly true. The condition you really need is not "along every line", but "along every path". Unfortunately the latter does not lend itself well to computation.
May 23, 2017 at 3:07	comment	added	Steven Gubkin		While I think it is good to discuss this idea, one should also point out that it doesn't always work: there are functions whose limit exists along every line passing through a point, but where the limit at that point does not exist.
May 23, 2017 at 1:31	history	answered	mweiss	CC BY-SA 3.0