Question on convergence of an ODE solution

Question

If $|\dot{x}(t)|$ converges to zero and $t|\dot{x}(t)|^4$ converges to zero, does it follow that $t|\dot{x}(t)|^2$ also converges to zero?

Answer 1

The fact that the function is some $\dot{x}$ doesn't really matter. The point is that $t x^2=\frac{tx^4}{x^2}$ is something going to zero divided by something going to zero, so what it does is indeterminate, depending on the rates the two pieces go to zero.

To get an example, look at $x=t^\alpha$. The original conditions are satisfied if $\alpha<-1/4$. Then $tx^2=t^{1+2\alpha}$ can still fail to converge if $\alpha>-1/2$.