Operational properties of equalities
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Operational properties of equalities in limits functions

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I would like to review two mathematical concepts that I understand are different but that sometimes seem to be confused, especially in cases of functions with limits to infinity.
These concepts are the fraction and the ratio.
The fraction can be considered as an eminently physical concept, prior to numerical mathematics and that even primitive humans should already know how to use.
For example, it might be easy for our ancestors to divide a food into equal portions to distribute among their families.

For the ratio the question is already different and we must know some mathematics, how to use a system like the decimal, to carry out this division operation.
But this division is eminently mathematical and numerical because with the decimal ratio, we can reach extremely sophisticated or extensive theoretical partitions with an approximation of a huge number of decimal places.
But of course, these ratios sometimes have their drawbacks.
The main one of them is that sometimes the ratio (or decimal division) does not allow us the translation or total transportation from fraction to ratio.
And if after doing the ratio or division, it result we have not achieved the full distribution of the dividend, and if now we want to join or add all the portions, it will not give us the full dividend because to these parts we must add the remainder of the division.

I have put in the drawing the fractional partition of a cheese, and also its adjustment as a ratio or decimal division.
In which we see that while the fractions are added and they give us the complete cheese again, in the ratio or decimal division, in addition to the dividends or decimal portions, we also have to add the remainder that remains from the division.
With this, and as I understand the question, I can agree that it is considered by convention or agreement that in certain limits such as the one in the example, we can accept the limits of certain sequences as correct, but that mathematical purity is not, because of these limits never can be gotten really.