Determine if the function is uniformly continuous

Question

Determine if the function: $$f(x)=\frac{x}{(1-x)^2}$$ $\forall x>1$ is uniformly continuous.

I know that it is not. But I'm having trouble proving it. I reach to a point that i get this: $$\frac{|x-y||1-xy|}{(1-x)^2(1-y)^2} \le \delta|1-xy|$$ and I cannot continue.

Any help would be appreciated!

Answer 1

You know that $\lim_{x\to1^+}f(x)=\infty$. So, for any $\delta>0$, you can find $x,y\in(1,1+\delta)$ such that $\bigl\lvert f(x)-f(y)\bigr\rvert\geqslant1$. But $\lvert x-y\rvert<\delta$.