Limit of $f(x,y) = \frac{y}{x}$ on restricted domain

Question

I know that this function
$$f(x,y) = \frac y x$$ does not have a limit at $(x,y) = (0,0)$ since, when we switch to polar coordinates, the function becomes
$$f(r, \varphi) = \tan(\varphi)$$ which does not depend on $r$, therefore the limit depends on the choice of the path.

However, what happens if we restrict the domain? for example:
$f: D \to \mathbb R^2 $ where $(x,y)\in D \iff |y| < |x^5|$
Now, is the limit likely to change right now? This restriction leads us to another one:
$$|r \sin(\varphi)| \le |r^5 \cos^5(\varphi)|$$ $$|\sin(\varphi)| \le r^4 |\cos^5(\varphi)|$$ Also, $$-r^4|\cos^5(\varphi)|\le \sin(\varphi) \le r^4|cos^5(\varphi)|$$ Therefore, $$\frac{-r^4|\cos^5(\varphi)|}{\cos(\varphi)} \le f(r, \varphi) \le \frac{r^4|\cos^5(\varphi)}{\cos(\varphi)}$$ Finally, when we consider the cases when $cos(x)$ is positive and negative, we get that this limit is in fact $0$.

Is it possible that the limit changes or have I committed some careless mistakes?

Answer 1

Yes in that restriction we have a limit, indeed note that

$$0<\left|\frac y x\right|<\frac{|x^5|}{|x|}=x^4\to 0$$