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The limit does not exist.
The expression is:
Check two paths:
Along :
\frac{xy}{x+y} = \frac{x^2}{2x} = \frac{x}{2} \to 0
Along :
\frac{xy}{x+y} = \frac{x(-x+x^2)}{x+(-x+x^2)} = \frac{-x^2+x^3}{x^2} = -1+x \to -1
Since the limit gives different values along different paths, the two-variable limit does not exist.
The expression is:
Check two paths:
Along :
\frac{xy}{x+y} = \frac{x^2}{2x} = \frac{x}{2} \to 0
Along :
\frac{xy}{x+y} = \frac{x(-x+x^2)}{x+(-x+x^2)} = \frac{-x^2+x^3}{x^2} = -1+x \to -1
Since the limit gives different values along different paths, the two-variable limit does not exist.
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