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**orir** i need to show that $\displaystyle f(x)=\frac{x^{2}}{1+x}$ is uniformly continuous at $\displaystyle [0,\infty) $.

i was trying to show that $\displaystyle \forall(\varepsilon>0)\exists(\delta>0)\forall|x-y|<\delta$ maintains $\displaystyle |\frac{x^{2}}{1+x}-\frac{y^{2}}{1+y}|\leq\varepsilon

$. $\displaystyle \:$what i was getting is that in the given range - $\displaystyle |\frac{x^{2}}{1+x}-\frac{y^{2}}{1+y}|\leq|x-y|\cdot[x+y+xy]\leq\delta\cdot[x+y+xy]$, and then i was stuck... how can i proceed from here?