**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](https://www.wolframalpha.com/input?i=plot+t%2F2*log%28t%5E2%29+from+t%3D-0.5+to+0.5):
$$x(t)=\frac{t}{2}\log(t^2)$$
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](https://i.stack.imgur.com/mdKzA.png)](https://i.stack.imgur.com/mdKzA.png)

From the plot it look like something quite feasible to be done, but if I focus on their [derivative](https://www.wolframalpha.com/input?i=D%5Bt%2F2*log%28t%5E2%29%2Ct%5D+) 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?**

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**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](https://youtu.be/_ckfHYuAjRA) 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](https://youtu.be/XXcWQWO-uHg) 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](https://math.stackexchange.com/q/4417566/909869), 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](https://youtu.be/JTvcpdfGUtQ).