Prove that for a function repeated Limit exist but simultaneous Limit does Not exist.

Question

Show that for the function defined by:

$f(x,y) = \begin{cases} 1 & xy =0\\ 0 & xy \ne 0\\ \end{cases}$

repeated limit exist at the origin but simultaneous limit does Not exist.

Now for repeated limit I can say that

$\displaystyle \lim_{x \to 0}( \lim_{y \to 0} f(x,y)) = \displaystyle \lim_{x \to 0}1 = 1$

Also

$\displaystyle \lim_{y \to 0}( \lim_{x \to 0} f(x,y)) = \displaystyle \lim_{y \to 0}1 = 1$ so both limit exist and are equal.

But I am not sure how to prove the second part ie Simultaneous limit do not Exist

Can anyone help me in this case ?

Thank you.

Answer 1

It follows from what you did that if the limit $\lim_{(x,y)\to(0,0)}f(x,y)$ existed, then it would have to be $0$. But you also have $\bigl(\forall x\in(0,\infty)\bigr):f(x,0)=1$ and therefore the limit $\lim_{(x,y)\to(0,0)}f(x,y)$ cannot be $0$.

Answer 2

We have that for $y=0$

$$\lim_{(x,y)\to (0,0)} f(x,y)=\lim_{(x,0)\to (0,0)} f(x,y)=1$$

but for $x=y=t \to 0$

$$\lim_{(x,y)\to (0,0)} f(x,y)=\lim_{t\to 0} f(t,t)=0$$