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I am given the problem $$ \lim_{(x,y) \to (0,0)}\frac{ y\sin^2{x}}{x^4 + y^2} $$ I know the first path to choose is either the x-axis or y-axis. This gives the limit of 0. Now I don't know how to choose the second path to prove that it does not exist.

Please forgive me for the format, I am new and I haven't learned it yet. Also sorry for my English if it's not properly correct.

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  • $\begingroup$ The answers give examples of paths that do this. As for how to find such examples, typically one first tries linear approaches, namely $y = ax$ where $a$ is a constant, and see if the expression is constant that varies with $a.$ If this doesn't work (i.e. you get a constant value for the limit that doesn't vary with $a),$ then try quadratic approaches, namely $y = x^2$ (or more generally, $y = ax^2$ where $a$ is a constant), or cubic approaches, or $n$'th degree approaches (i.e. $y = ax^n$ where both $a$ and $n$ are constants). (continued) $\endgroup$ – Dave L. Renfro Aug 19 at 10:44
  • $\begingroup$ Here is an example where you need to go beyond "polynomial approaches". $\endgroup$ – Dave L. Renfro Aug 19 at 10:45
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Hint: Take $y=x^{2}$ and show that the limit is $\frac 1 2$. Use the fact that $\frac {\sin \, x } x \to 1$ as $x \to 0$.

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  • $\begingroup$ Oh that makes sense! I forgot about that theorem. Thank you! $\endgroup$ – avatar-korra Aug 19 at 9:50
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If $f(x,y)=\frac{y \sin^2 x}{x^4+y^2}$, then compute

$\lim_{x \to 0}f(x,0)$ and $\lim_{x \to 0}f(x,x^2)$.

Consequence ?

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