I am trying to solve the following limit and Wolfram Alpha says that it does not exist, but I am not able to prove it.
$$
\lim_{(x, y) \to (0,0)} \frac{\sin(xy)}{x^2+y^2}
$$
I have tried to move toward the origin on $x=0$ and I get:
$$
\lim_{y\to0} \frac{\sin(0)}{y^2}=0
$$
The same if I move toward the origin on $y=0$:
$$
\lim_{x\to0} \frac{\sin(0)}{x^2}=0
$$
Then I tried to move on $x=y$:
$$
\lim_{y\to0} \frac{\sin(y^2)}{2y^2}=0
$$
I can't spot how to prove that there is another value of the index which is not $0$, but I am not able to get rid of the numerator.
Best Answer
Your last limit is wrong. By substituting $x=y^2$, $$\lim_{y\to 0}\frac{\sin y^2}{2y^2} = \frac12 \lim_{x\to0}\frac{\sin x}{x} = \frac12 $$