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