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Here is a problem in calculus:

Let $ f(x,y)= \begin{cases} y+x\sin\bigg(\frac{1}{y}\bigg),& y\neq0\\\\0,& y=0 \end{cases}$ show that :

  1. $\lim_{(x,y)\to (0,0)} f(x,y)$ and $\lim_{y\to 0} \big(\lim_{x\to0} f(x,y)\big)$ exist.

  2. $\lim_{x\to 0} \big(\lim_{y\to0} f(x,y)\big)$ doesn't exist.

It is an elementary question here and I could solve the first part only. Taking different paths couldn't also help me. May I ask to help me about the second part? Give me the right path reaching the origin. Thanks.

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The fact that the limit $\lim_{x\to 0}\lim_{y\to 0}f(x,y)$ doesn't exist is due to the fact that $\lim_{y\to 0}\sin\frac 1y$ doesn't exist. To see that, work with the sequences $y_n:=\frac 1{2\pi n}$ and $z_n:=\left(2\pi n+\frac{\pi}2\right)^{—1}$.

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    Thanks a lot. I am searching for this magic seq. Thanks Davide. :)2012-09-20
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$\lim \limits_{x\to 0} \left(\lim \limits_{y\to 0} f(x,y)\right)$ does not exist, because for $x\ne 0$ doesn't exist $\lim \limits_{y\to 0} \sin{\frac{1}{y}}$