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.