TITLE: Wrong Wolfram|Alpha limit? $ f(x,y) = \frac {xy}{|x|+|y|} $ for $(x,y)\to(0,0)$
QUESTION [23 upvotes]: 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?

REPLY [2 votes]: Any function $f(x,y)$ such that $|f(x,y)|\le |x|$ has limit $0$ at $(0,0).$ And we have that here:
$$\left | \frac{xy}{|x|+|y|} \right | = |x|\left | \frac{|y|}{|x|+|y|} \right | \le |x|.$$