I have a question. I want to check whether the following limit exists:  $$\lim_{(x,y) \rightarrow (0,0)} \frac{xy}{|x|+|y|}.$$   I have already searched on the internet and they say many different things. In my own book there stood that it doesn't exist. Is that true? And how do you show that it doesn't or does exist by this limit?  Thank you
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.  . Am I wrong or is Wolfram|Alpha wrong?
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