let $$f(x)= (x+y) \sin(\frac{1}{x}+ \frac{1}{y}), x\neq0, y\neq0$$ else $f(x,y)=0$. I need to show that $f(x,y)$ is continuous at $(0,0)$ and that $\lim_{x\to0}f(x,y)$ for $y\neq0$ exists.
Continuity at $(0,0)$ is easily shown by observing that $|f(x,y)|<|x+y|$. However, I think that the second part is incorrect and the asked limit doesn't esists. (as there will be $\sin(\infty)$ in the expression.) Can please someone confirm if thats correct or show how the $\lim_{x\to0}f(x,y)$ for $y\neq0$ exists.
Your arguments for the continuity at $(0,0)$ are o.k.
Indeed , the second part is incorrect. Let $y \ne 0$ and define
$$x_n:=\frac{y}{y n \frac{\pi}{2}-1}$$
for $n$ so large such that $y n \frac{\pi}{2}-1>0.$ Then $x_n \to 0$, but
$f(x_n,y)=(x_n+y) \sin (n \frac{\pi}{2})$ does not converge.