Why is my solution incorrect? I was trying to compute this limit:
$$\lim_{x \to 0}\lim_{y \to 0} (x+y)\sin{\frac{x}{y}}$$
And this is my solution:
$$\lim_{x \to 0}\lim_{y \to 0}|(x+y)\sin{\frac{x}{y}}|\leq\lim_{x \to 0}\lim_{y \to 0} |(x+y)|=0$$
So I got the limit 0.
The answer was different. I have no idea what is wrong with my solution?
 A: Note that
$$
\lim_{y\to0}(x+y)\sin\left(\frac xy\right)
$$
is indeterminate for each non-zero $x$. Thus, there is no way to compute
$$
\lim_{x\to0}\lim_{y\to0}(x+y)\sin\left(\frac xy\right)
$$
Whereas,
$$
\lim_{x\to0}(x+y)\sin\left(\frac xy\right)=0
$$
for each non-zero $y$, so
$$
\lim_{y\to0}\lim_{x\to0}(x+y)\sin\left(\frac xy\right)=0
$$

Note that
$$
\lim_{\substack{(x,y)\to(0,0)\\y\ne0}}(x+y)\sin\left(\frac xy\right)=0
$$
Thus, on any path toward $(0,0)$ where $y\ne0$, the limit will be $0$. However, because
$$
\lim\limits_{y\to0}(x+y)\sin\left(\frac xy\right)
$$
does not exist for any $x\ne0$, the function cannot be extended continuously to the line $y=0$ and this messes up
$$
\lim_{x\to0}\lim_{y\to0}(x+y)\sin\left(\frac xy\right)
$$
A: You are intuitively trying to use a generalization of the following result from calculus of single variable : if $|f(x) |\leq g(x) $ as $x\to a$ and $g(x) \to 0$ as $x\to a$ then $f(x) \to 0$ as $x\to a$. This holds for limits of functions of two variables also. But you are dealing with an iterated limit and not a double limit.
A double limit always involves both variables moving together towards the point under consideration. An iterated limit does not work in this fashion and it is possible that double limit exists but the iterated limits fail to exist.
This is what happens in the current question. The double limit of the function is $(x+y) \sin(x/y) $ is $0$ precisely because of the inequality you have used. But the iterated limit $\lim_{x\to 0}\lim_{y\to 0}$ does not exist. The inequality $|(x+y) \sin(x/y) |\leq |x+y| $ is fine but when you take limits as $y\to 0$ then you can see that the RHS tends to $|x|$ which may or may not be $0$. And hence there is no guarantee that the limit of LHS exists. If you change the function to $xy\sin(x/y) $ then you don't face this problem and both the iterated limits as well as the double limit are $0$.
Iterated limits are nothing but single variable limits applied one after another. And their theory is much simpler. The confusion here comes because you are trying to use iterated limits but side by side also trying not to treat the variables independently. 

Note: The definition of double limit assumes that the point under consideration (here $(0,0)$) is a point of accumulation of the domain of the function.
