Show that $\lim_{(x, y) \to (0, 0)} \frac{y + \sin x}{x + \sin y}$ does not exist.

Let $y=-x$.

Thus, $$\lim_{x\rightarrow0}\frac{-x+\sin{x}}{x-\sin{x}}=-1.$$

But for $y=x$ we obtain: $$\lim_{x\rightarrow0}\frac{x+\sin{x}}{x+\sin{x}}=1.$$

If you go along the path $y=-x$

$$\frac{-x+\sin x}{x+\sin(-x)} = -1$$

therefore the limit does not exist.