$f(x,y) = \frac{x y^2}{x^2 + y^4}$ with $(x,y) \in \mathbb{R}^2 \backslash \{0,0\}$

$f(x,y)$ is not defined in $(0,0)$. We are going to look at $\lim_{(x,y) \to (0,0)}f(x,y)$.

a) First determine the value of $f(x,y)$ for any random straight line through $(0,0)$. Determine the limit of $f(x,y)$ if $(x,y)$ goes to $(0,0)$ through this line.

b) Now determine the behavior of $f(x,y)$ on $x=y^2$ in the neighborhood of $(0,0)$

c) Proof that $$\lim_{(x,y) \to (0,0)} f(x,y) \neq 0$$

d) Now proof that the limit doesn't exist at all.

So I started with the first exercise and used a general expression for the straight lines through $(0,0)$. So I said $y=ax$ with $a \in \mathbb{R}$ and try to rewrite the function in the following way:

$$f(x,ax)=\frac{a^2 x^3}{x^2 + a^4x^4}=\frac{a^2 x}{1+ a^4x^2}$$ and taking the limit I thus get $\lim_{x \to 0} f(x,ax) = 0 $.
Is this the correct approach?

Because in a similar way I find the following limit allong $x=y^2$.

$$\lim_{y \to 0} f(y^2,y) = \lim_{y \to 0} \frac{y^4}{y^4+y^4} = \lim_{y \to 0} \frac{1}{2} = \frac{1}{2}$$

For questions c and d I don't know how to write this down properly. Obviously from what I have done I see that the limit doesn't exist but I'm stuck at this point. I don't want to get the answer but rather a tip on how to start so that I can figure the rest out by myself.

  • $\begingroup$ For (c), what would $\displaystyle \lim_{(x,y) \to (0,0)} f(x,y) = 0$ mean, for example in epsilon-delta terms? Then show that that is not true for this $f(x,y)$. $\endgroup$ – Henry May 16 '15 at 21:47

(a) and (b) looks good.

For (c) consider the sequence $(x_n, y_n) := \left( \frac{1}{n^2}, \frac 1 n \right)$ for $n \in \mathbb N^\times$. Then we have $\displaystyle \lim_{n \to \infty} (x_n, y_n) = (0,0)$. Have a look at $\displaystyle \lim_{n \to \infty} f(x_n, y_n)$ ((b) gives you the answer what this limit is), and conclude that $\displaystyle\lim_{(x,y) \to (0,0)} f(x,y) \neq 0$.

For (d) do the same with $(x_n, y_n) := \left( \frac 1 n , \frac 1 n \right)$. What is $\displaystyle \lim_{n \to \infty} f(x_n, y_n)$ in this case?


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