Quick question about two-path limit test

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October 13th 2010, 09:17 PM
downthesun01

Quick question about two-path limit test

How do you decide what two paths to take? For example:

$\frac{x^{2}y}{x^{4}+y^{2}}$

Can I pick any paths I want? For example:

The line $x=0$

$lim_{x\rightarrow 0}$ $\frac{0}{y^{2}}=0$

And the line $y=x^{2}$

$lim_{y\rightarrow x^{2}}$ $\frac{x^{2}*x^{2}}{x^{4}+x^{4}}=\frac{1}{2}$
October 14th 2010, 12:48 AM
Prove It

What point are you making this function tend to?
October 14th 2010, 12:50 AM
downthesun01

Sorry. (0,0)
October 14th 2010, 01:00 AM
Prove It

Quote:

Originally Posted by downthesun01

How do you decide what two paths to take? For example:

$\frac{x^{2}y}{x^{4}+y^{2}}$

Can I pick any paths I want? For example:

The line $x=0$

$lim_{x\rightarrow 0}$ $\frac{0}{y^{2}}=0$

And the line $y=x^{2}$

$lim_{y\rightarrow x^{2}}$ $\frac{x^{2}*x^{2}}{x^{4}+x^{4}}=\frac{1}{2}$

This is probably easiest to solve by converting to polars, remembering that $x = r\cos{\theta}, y = r\sin{\theta}, x^2+y^2=r^2$ ...

$\displaystyle{\lim_{(x,y) \to (0,0)}\frac{x^2y}{x^4 + y^2} = \lim_{r \to 0}\frac{r^2\cos^2{\theta}\cdot r\sin{\theta}}{r^4\cos^4{\theta} + r^2\sin^2{\theta}}}$

$\displaystyle{= \lim_{r \to 0}\frac{r^3\cos^2{\theta}\sin{\theta}}{r^2(r^2\cos ^4{\theta} + \sin^2{\theta})}}$

$\displaystyle{= \lim_{r\to 0}\frac{r\cos^2{\theta}\sin{\theta}}{r^2\cos^4{\th eta} + \sin^2{\theta}}}$

$\displaystyle{= \frac{0}{\sin^2{\theta}}}$ .

Since this value could change when $\sin^2{\theta} = 0$ , we can say that the limit does not exist.
October 14th 2010, 01:14 AM
downthesun01

Thanks but we haven't learned about polars yet. If someone could explain it for me by using the two-path method, I would much appreciate it. Thanks
October 14th 2010, 04:12 AM
HallsofIvy

The original question was "how do you decide which two paths to take". The answer is, you don't. You try different paths, perhaps many, until you find two which give different limits. If there exist different paths that give different limits, then the limit itself does not exist.

The problem with that is that you can't try all paths and there is no way of knowing in advance which, if any, paths will give different limits. Even if you tried all straight lines, by using "y= mx" with m left variable, and found that you got the same limit for all straight line paths, that is not enough. There are functions which give the same limit for all straight paths but different limits for curved paths and so do NOT have a "limit".

If, for some paths, you get different limits then you know that the limit itself does not exist. If, however, you get the same limit for every path you try, you still do NOT know whether or not the limit itself exists.