# Determining existence of limit with multiple variables: $\lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3}$

Given the following limit:

$$\lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3}$$

And the instrucion to "Determine whether the limit exists, give a complete argument", would the following be a "complete argument"?

Approaching the limit from the line y=0, gives $\lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} = \lim_{(x,y)\to (0,0)} \frac{0}{x^3+y^3} = 0$

Approaching the limit from the line y=x, gives $\lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3} = \lim_{(x,y)\to (0,0)} \frac{x^3}{2x^3} = \frac{1}{2}$

These limits do not agree, thus the original limit $$\lim_{(x,y)\to (0,0)} \frac{xy^2}{x^3+y^3}$$ does not exist.

Or should another method besides approaching from different lines be used to give a "complete argument" whether this limit exists be given?

• It is okay. You don't need anything else :) – Ant Mar 11 '15 at 13:43
• – Arnaud D. Dec 3 at 18:13

In fact, the limit $\lim_{(x,y)\to (a,b)}f(x,y)$ exists and equals some value like $c$ if and only for any sequence $\{x_n\}_{n}$ converging to $a$ and any sequence $\{y_n\}_{n}$ converging to $b$ one has that $$\lim_{n\to \infty}f(x_n,y_n)=c.$$
So for proving of non-existence of $\lim_{(x,y)\to (a,b)}f(x,y)$, it is enough to list two sequences $\{(x_n,y_n)\}_{n}$ and $\{(x'_n,y'_n)\}_{n}$ both converging to $(a,b)$ while $$\lim_{n\to \infty}f(x_n,y_n)\ne\lim_{n\to \infty}f(x'_n,y'_n).$$
Note that here you have almost done the same by mentioning $x_n=a_{n}$ and $y_n=0$, and also $x'_n=a_{n}$ and $y'_n=a_{n}$, where $\{a_n\}_n$ is an arbitrary sequence such that $a_{n}\to 0$, e.g. like $a_n=\frac{1}{n}$