Show that the limit
$$\lim_{(x, y) \to (0, 0)} \dfrac{y + \sin x}{x + \sin y}$$
does not exist. I tried using two-path test but all of them gave the same value $1$. I tried using paths $y = 0, y = kx, y = \sin x$ but all of them give limit $1$. Since this is the only method taught as of now, I would like to know how I can use two-path test to show this.
Best Answer
Let $y=-x$.
Thus, $$\lim_{x\rightarrow0}\frac{-x+\sin{x}}{x-\sin{x}}=-1.$$
But for $y=x$ we obtain: $$\lim_{x\rightarrow0}\frac{x+\sin{x}}{x+\sin{x}}=1.$$