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I think I have been exposed since years ago to this line of reasoning:

if $ v\to c $, then $ \Delta t \to \infty $. As $\displaystyle v=\frac{\Delta s}{\Delta t} $, it's like a natural reaction to some massive object approaching light speed in order to prevent $v=c$.

Similarly, if $v \to c$, then $m \to \infty$. As $ F=ma$, accelerating the object needs more and more force, so that $c$ is ungraspable.

Is this thinking correct or simplistic and even worse? Is there, anyway, an analogous explanation of length contraction?

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If those thoughts help you to remember time dilation and relativistic energy increase that's fine. But once you have time dilation, you get space/length contraction to preserve light speed. After all if someone's clock is running slow, they better measure distances as smaller too or else when they see some light shoot by they aren't going to think it was travelling at c.

So if a lightspeed limit helps you remember those two effects, then also remember that different observers agree on the speed of light to get that space/length contraction to "hide" the time dilation.

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As far as I know relativistic mass increase is a concept long abandoned as an interpretation (as long ago as Einstein). Instead the approach is to use relativistic (3-) momentum where the mass appears only as the rest mass and it is invariant under Lorentz transformation.

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