Limit multivariable calculus I want to find the limit:
\begin{align}
    \lim_{(x,y)\mapsto(0,0)} (x+y)\sin(\frac{1}{x})\sin(\frac{1}{y}).
\end{align}
I have plotted it and evidently it does not exist. I have tried to prove it by finding two different trajectories where the limit is different, but I have not managed to do so. I think it is probably with logarithmic or exponential functions, or maybe it is much simpler than that. I would appreciate some help.

HINT
As suggested by @peek-a-boo, you can proceed as follows:
\begin{align*}
0 < \left|(x+y)\sin\left(\frac{1}{x}\right)\sin\left(\frac{1}{y}\right)\right|\leq |x + y|
\end{align*}
Now you can apply the squeeze theorem as $(x,y)\to(0,0)$.
Can you take it from here?
