Decide whether the function $f(x,y)=\lim_{(x,y) \to (0,0)} \frac{x^4+y}{x^8 + y^2}$ has a limit at $(0,0)$ Let $f:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ be the function defined as follows: 
$$f(x,y)=\frac{x^4+y}{x^8 + y^2}$$
I want to compute the limit of $f$ at $(0,0)$
$$\lim_{(x,y) \to (0,0)}f(x,y)=\lim_{(x,y) \to (0,0)} \frac{x^4+y}{x^8 + y^2}$$
This is my attempt..
If $y=x^4$ then, 
$$f(x,x^4)= \frac{1}{x^4}$$
Therefore the limit does not exist.

Your attempt could use a little more explanation, but in essence, it is OK.

For example, you could explain that you already know that if the limit exists, and because $g: x\mapsto (x,x^4)$ is a continuous function, then the limit $$\lim_{x\to 0} f(g(x))$$ should also exist, but since it does not, this leads to a contradiction.
Or, you could say that you proved that for every $\epsilon>0, M > 0$, there exist some $(x,y)$ such that $\|(x,y)\|\leq \epsilon$ and $f(x,y)>M$, and therefore, $f$ is not bounded on any neighborhood of $(0,0)$ and a limit cannot exist.
