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I have an example from my prof. He said, in evaluating limits, we can use L1 (along x-axis), L2 (along y-axis), and L3 (y = kx). After that, if the limits come to the same finite number, then the limitmightexist. Here's the example.

[tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^3y}{x^6+y^2}[/tex]

The L1, L2, L3 limits all come to zero.

Then being the prof because he is smart, he used L4 = y = x^3, and showed the limit equals some other number, which proved the limit doesn't exist.

[tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow x^3}} f(x,y)=\frac{x^3y}{x^6+y^2}=1/2[/tex]

How can I, as a student, come up with some "random" function such as [itex]y=x^3[/itex] to show that the limit does not exist??

How do I know if the limit existfor sure?? Is there a way to tell??

Thanks for the help!

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# Homework Help: LIMITS - Multi-variable calculus

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