How to find the all possible values of an undefined limit?

Question

The following limit is not defined: $$ \lim_{(x,y)\rightarrow (0,0)} \frac{xy}{x^2(1+y)}.$$ However, I am interested in a way to find all possible values as $(x,y)$ approaches $(0,0)$. My intuition is that the answer are $\pm\infty$ and $0$. And the method of approaching this is probably by representing $x,y$ as complex number. But I am unsure of how to proceed. Any hint or reference would be much appreciated.

Answer 1

Take $(x,y)=(t,kt),$ where $k\in\mathbb R$ and $t\rightarrow0$.

Also, take $(t^2,t)$.

Answer 2

Let $y=x$, then $$\frac{x^2}{x^2+x^3}=\frac{1}{1+x}$$ as $x$ tends to $0$, the limit $f(x,y)$ converges to $1$.

For the second limit, let $y=x^2$, then $$f(x,y)=\frac{y^{3/2}}{y+y^2}$$ which simplifies to:

$$\frac{y^{1/2}}{1+y}$$ and as you let $y$ tend to $0$ the function converges to $0$

Therefore the limit does not exist.

Answer 3

Ignore the factor $1+y$, which plays no role, and simplify the $x$. Now

$$\frac yx$$ can take any value.