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Is possible to achieve an infinite speed in real life?

I know beforehand that relativity theory forbids anything from travel faster than speedlight, so please stay with reading the full question first before answer - is more a conceptual/mathematical issue

For generality I will work with dimensionless units.

Imagine a point-particle system which position in time is described by the equation: $$x(t)=\frac{t}{2}\log(t^2)$$ and the particle starts moving at time $t_0=-0.2$ with initial speed and the experiments ends measuring its position at time $t_f=0.2$ following the equation $x(t)$. position_vs_time

From the plot it look like something quite feasible to be done, but if I focus on their derivative I will note that there is a point in space where it maximum speed will rise to infinity without any discontinuity on the position of the particle: $$\frac{d}{dt}x(t) = 1+\frac{\log(t^2)}{2}$$

Since the maximum speed diverges $\lim\limits_{t\to 0}\|\dot{x}(t)\|_\infty \to \infty$:

  1. Is this system attainable under current interpretation of physics laws?
  2. Where I am messing up with the concept of speed and its divergence?
  3. If this is an achievable system, Do you know a mechanic system that behaves like this?

Motivation (Added later)

Since I am receiving some downvotes, I would like to made an explanation of why this question is interesting:

In this video I found by accident on youtube, Terence Tao, one of the most important nowadays mathematician, is reviewing finite-time blow up solutions to differential equations. There, he talk about the possibility of having this solutions which reaches infinity in finite time for the Navier-Stokes equation, which is indeed a widely known physical system.

Thinking in the speed limits rise in Relativity due to Einstein, I was wondering if this talk is just a mathematical curiosity which cannot be happening in real life, but then I saw this video about the Euler´s disk toy, where, for the wobbling rate the solutions indeed shows to have a finite time blow up (it just a frequency ratio so inertia restrictions indeed are not applying).

On this question, also I figure out that sometimes a finite time blow up could be thought as the reciprocal of a solution of finite duration to a differential equation (meaning here, it achieves zero by it own dynamics and stays there forever after).

So if some people are researching this kind of things, I was thinking that maybe something that achieving the blow up in their speed profile could also be possible (at least is possible in the math).... this is why I keep unitless/dimensionless the differential equations, since maybe a model where inertia is not involved could fit the presented situation.

Another example, the point particle could be the displacement of a shadow, as the example shown on this video.

Dale
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Joako
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    Note infinite speed implies infinite kinetic energy which s clearly unphysical. – StephenG - Help Ukraine Apr 04 '22 at 22:09
  • @StephenG-HelpUkraine You are right, but it only applies if inertia is directly involved, as example, the wobbling ration of the Euler's Disk toy indeed shows a finite-time blow up behavior (see here)... this is why I keep dimensionless the function, to keep it open to every possible model. – Joako Apr 04 '22 at 22:29
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  • You can't just write down any function $x(t)$ and call it a "physical system". The trajectories $x(t)$ of a physical system are derived as solutions to its equations of motion, not postulated a priori. 2. If your question is just whether a function of the type $t\mathrm{log}(t^2)$ occurs anywhere in physics, your question is much too broad/unfocused to be on-topic here. 3. It is completely unclear what you mean by infinite speed being "achievable" when you simultaneously seem to say what we can ignore relativity.
    • – ACuriousMind Apr 04 '22 at 23:11
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    It is not appropriate to modify a question in a way that invalidates responses you have already received to the original question. I hav reverted the inappropriate edit – Dale Apr 05 '22 at 00:19
  • @Dale I did it because it was closed and the comment was to made the question more specific. So if I can´t change it without violating the site standards I will delete it and ask the proper version again. Thank for taking the time to edit it. – Joako Apr 05 '22 at 02:04
  • @Dale I will post the new version, but the system doesn't allow me to delete this one, so If you are still seen it it is because of that. – Joako Apr 05 '22 at 02:43