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It is noteworthy that one cannot simply divide any length more than the Planck-length. If so, one cannot simply divide any volume more than the $(Planck-length)^3$.

So if I want to find the limit of some $n$,

$${(\frac {a}{a+V/n})}^{n-1}$$

to infinity.

where $a$ is a constant and $V$ is a volume,

it is not correct to limit $n$ to infinity right?

because when limiting to infinity I surpass the plank volume.

So am I violating any laws of physics if I'm limiting this expression to infinity?

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    $\begingroup$ You make the rules when doing mathematics. If you want to take the limit of the volume going to zero, do it. If you think it makes sense to stop at the Planck volume, do that. The laws of physics apply to (observations of) reality, not to mathematical reasoning. $\endgroup$ – vosov Nov 28 '15 at 12:11
  • $\begingroup$ "It is noteworthy that one cannot simply divide any length more than the Planck-length" This presumes that the plank-length represents a discretation length, a position for which there is absolutely no evidence either way at this point. $\endgroup$ – dmckee Nov 29 '15 at 0:56
  • $\begingroup$ @vosov is there a way to stop the limit at the plank-length? $\endgroup$ – slhulk Nov 29 '15 at 4:45
  • $\begingroup$ @dmckee I'm sorry I don't get you correctly.. are u saying there are no evidences for the existence of the idea known as the plank-length? $\endgroup$ – slhulk Nov 29 '15 at 4:45
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    $\begingroup$ The only thing that is known about the plank length is that it represents a final lower limit on the measurable distance between two points, but you should not interpret that as meaning that space necessarily has a grid size. There is no reason to expect that, and no evidence that it is so. Indeed, the conservation of angular momentum suggest it is not so. There is nothing that prevents space being continuous even though there is a plank-length. For instance if there is a grid than no measurement could ever return 1.5 plank-lengths between two points, but if space is continuous it could. $\endgroup$ – dmckee Nov 29 '15 at 5:13
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We can only speculate about what happens at the Planck length, we are a long way from any experiment that might probe into those length scales.

This has virtually no effect on any calculations we ever do however. As such it is not something worth worrying about when trying to solve a problem which is at scales we encounter in day to day life. In your example you will see that if you plug in any realistic volumes for $a$ and $V$ and choose $n$ such that $V/n$ is a planck length cubed you will get incredibly close to the limiting value $\exp(-V/a)$.

In short, it may well be that sometimes our sums, limits or integrals should be cutoff by the planck scale however this scale is so incredibly huge compared to everything else we know that this nearly never needs to be a consideration.

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  • $\begingroup$ you are implying that an error of which the magnitude $10^{-34}$ (in the metric system) is not considerable? $\endgroup$ – slhulk Nov 29 '15 at 4:49

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