Calculus – Wrong Wolfram|Alpha Limit? f(x,y) = xy / (|x| + |y|) as (x,y) ? (0,0)

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I have this function:

$$ f(x,y) = \frac {xy}{|x|+|y|} $$

And I want to evaluate it's limit when $$ (x,y) \to (0,0)$$ My guess is that it tends to zero. So, by definition, if:

$$
\forall \varepsilon \gt 0, \exists \delta \gt 0 \diagup \\ 0\lt||(x,y)||\lt \delta , \left|\frac{xy}{|x|+|y|}\right| \lt \varepsilon
$$
Then
$$
\lim_{(x,y)\to(0,0)}\frac {xy}{|x|+|y|} = 0
$$
So:

$$
\left|\frac{xy}{|x|+|y|}\right| = \frac{|xy|}{|x|+|y|}
= \frac{|x||y|}{|x|+|y|} \le 1 |y| \lt \delta
$$

So for any $$\delta \lt \varepsilon$$ the inequality is true. Hence, the limit exists and is equal to zero.

Wolfram|Alpha says that the limit does not exist. Am I wrong or is Wolfram|Alpha wrong?

Best Answer

Pretty simply, we have $$ |xy|=\max(|x|,|y|)\min(|x|,|y|)\tag{1} $$ and $$ |x|+|y|\ge2\min(|x|,|y|)\tag{2} $$ Therefore, $$ \left|\frac{xy}{|x|+|y|}\right|\le\frac{\max(|x|,|y|)}{2}\tag{3} $$ Thus, $$ \lim_{(x,y)\to(0,0)}\left|\frac{xy}{|x|+|y|}\right|\le\lim_{(x,y)\to(0,0)}\frac{\max(|x|,|y|)}{2}=0\tag{4} $$

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