Is 0 divided by a non-zero indeterminate equal to 0.

Question

Consider the following solution:

$$ \lim\limits_{(x, y) \to (0, 0)} \dfrac{xy^4}{x^4+y^4}$$

Divide both numerator and denominator by $y^4$

$$ = \lim\limits_{(x, y) \to (0, 0)} \dfrac{x}{\left(\dfrac{x}{y}\right)^4+1}$$

$$ = \dfrac{ \lim\limits_{(x, y) \to (0, 0)}x}{ \lim\limits_{(x, y) \to (0, 0)}\left(\dfrac{x}{y}\right)^4+1}$$

The numerator is 0 and the denominator is non-zero, hence the limit is 0

=============================

Are you satisfied with this solution, if not, why?

Edit: Thank You for the help guys.

My mistake was looking at the final limit and not paying attention to how I got to that stage, "by dividing both numerator and denominator by y^4", that step itself is not allowed because y can be 0.

The above solution will be complete if I include another case where $y=0$, because when you divide by $y^4$, you are implicitly stating that y is not 0.

Answer 1

What you have shown is that assuming $y\neq 0$ the limit is zero but it doesn't suffice.

What matter is that for $y\neq 0$

$$ \dfrac{|x|}{\left(\dfrac{x}{y}\right)^4+1} \le |x| \to 0$$

and since for $y=0 \implies \dfrac{xy^4}{x^4+y^4}=0$ the proof is complete.

Answer 2

There is an easy way: \begin{align*} \left|\dfrac{xy^{4}}{x^{4}+y^{4}}\right|&=|x|\dfrac{y^{4}}{x^{4}+y^{4}}\\ &\leq|x|\\ &\rightarrow 0. \end{align*}

Answer 3

One thing that has not been adressed by the existing answers is that even if you restrict your reasoning to the cases where $y\neq 0$, it's still wrong. To write something like $$\lim_{(x,y)\to (0,0)}\frac{f(x,y)}{g(x,y)}=\frac{\lim_{(x,y)\to (0,0)} f(x,y)}{\lim_{(x,y)\to (0,0)}g(x,y)}$$ you have to make sure that $g$ is non-zero around the identity and that both $\lim_{(x,y)\to (0,0)} f(x,y)$ and $\lim_{(x,y)\to (0,0)} g(x,y)$ exist, and that the latter is not $0$. But here $g(x,y)=\left(\frac{x}{y}\right)^4+1$ does not have a limit when $(x,y)\to (0,0)$, so you can't actually write something like $$ \lim\limits_{(x, y) \to (0, 0)} \dfrac{x}{\left(\dfrac{x}{y}\right)^4+1}= \dfrac{ \lim\limits_{(x, y) \to (0, 0)}x}{ \lim\limits_{(x, y) \to (0, 0)}\left(\dfrac{x}{y}\right)^4+1}$$ because the right-hand side does not make sense.