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Can anyone find the maximum and the minimum value of the following expression $$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}$$ for all $x,y,z$ belongs to $\mathbb{R}$

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    $\begingroup$ what kind of numbers are $$x,y,z$$? $\endgroup$ – Dr. Sonnhard Graubner Jun 24 '17 at 15:40
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The maximum does not exist. Try $x>-1$ and $xy+x+1\rightarrow0^+$.

The minimum does not exist. Try $x<-1$ and $xy+x+1\rightarrow0^+$.

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