Two variable limit

Question

Suppose I have a function which is defined in different parts, for example:

$$f(x,y)=y\cos\left(\frac{x}{y}\right)\ \ \ y\neq0$$ $$f(x,0)=0$$

and I have to calculate the limit when $(x,y)\rightarrow (x_0,0)$. Is it possible to calculate it separately through the curves $y=0$ and $y\neq 0$ and say that if both are the same, then the limit is that number? I have seen this procedure a number of times, but I have the following doubt: it would be possible to find a curve $C$ with points satisfying both $y=0$ and $y\neq0$, and therefore the division made above does not consider all the curves passing through $(x_0,0)$.

Which is the correct argumentation to solve a limit of this kind?

Answer 1

Using: $$\large-1\le\cos x\le1\\\implies\large-y\le y\cos\frac{x_0}{y}\le +y$$ Think what will happen if $y\to0$

Spoiler:Hint:

Think in terms of Sandwich theorem, maybe the limit is $0$? Is it?

Answer 2

When $(x,y)\rightarrow (x_0,0)$ I have $$ 0\leftarrow 0\leq |f(x,y)| \leq \max\{|ycos(\frac{x}{y})|,0\}\leq \max\{|y|,0\} \rightarrow 0 $$ Using the sandiwch rule I have that the limit is 0.

Thank you very much for your help.