Lipschitz and uniform continuity
Solution 1:
The basic idea is that around $0$, the change in $f(x)$ is not bounded by a constant times the amount that we change $x$ (note that the derivative becomes arbitrarily large). More formally, for any fixed $k>0$,
$$\sqrt{\frac{1}{(k+1)^2}} = \frac{1}{k+1} > k \frac{1}{(k+1)^2}$$
which violates the Lipschitz condition.