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In the calculus 3 textbook there is a problem that asks to show that $\lim_{(x,y) \rightarrow (0,0)} \frac{x+\sin y}{y+\sin x}$ does not exist. I found two paths $y=x$ and $y=-x$ approaching to zero with two different limits $1$ and $-1$. I also tried other polynomial paths like $y=x^m$ or $x=y^m$, and the limit is always equal to $1$. Hence even if $\lim_{(x,y) \rightarrow (0,0)} \frac{x+\sin y}{y+\sin x}$ does not exist, it is expected that in fact $\lim_{(x,y) \rightarrow (0,0)} |\frac{x+\sin y}{y+\sin x}|=1$. However I'm not able to prove this. Any comments or ideas would be really appreciated.

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    $\begingroup$ The problem is that within any $\delta$-disc around the origin, you can always find some point $(x, y)$ such that $x + \sin y = 0$ (and $y + \sin x \not= 0$), so the limit fails. $\endgroup$ – Brian Tung Oct 3 '19 at 21:44
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    $\begingroup$ @BrianTung, Thanks for this clever observation! $\endgroup$ – student Oct 3 '19 at 22:13
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Summing up what you and Brian Tung already wrote: If you approach $(0,0)$ along the path $x= -\sin y$ you get $\left| \frac{x+\sin y}{y+\sin x}\right| = \left| \frac0{y-\sin(\sin y)}\right|=0$. However if you take the path $x=y$ you get $\left| \frac{x+\sin y}{y+\sin x}\right| = \left| \frac{y+\sin y}{y+\sin y}\right| = 1$.

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