Does this "theorem" of limits hold in general?

[walking]

I recently found the centre of mass of a semicircle using polar coordinates, by first finding the centre of mass of a sector, and then summing all the sectors from 0 to pi to get the centre of mass of the semicircle. However, being a beginner at integrals, I struggled for a long time getting the wrong answer because at first, I was assuming the sectors could be modeled as straight lines with 0 area due to the angle being small. This led me to wrongly assume that the centre of mass of each sector was simply halfway along its radius, and by summing all these "sectors" from 0 to pi, I got the wrong answer. It was only when I assumed each sector actually "behaved" like a 2D sector, before simultaneously taking limits tending to 0 for the double integrals, that I was able to get the correct answer.

The lesson I took from this was that when dealing with limits, one should not take limits for one thing before taking limits for another thing: for example, in the semicircle problem, I took the limit of each sector as the angle tends to 0 (resulting in a straight line with 0 area), before taking the limit of the summation of all the sectors in the semicircle.

Is this a general theorem when dealing with limits, and if so, what is the theorem which deals with this? If there is not a specific theorem, what is the reasoning behind it? I am aware that it may involve analysis (which I haven't studied yet), but I simply want to know if there is a rigorous reasoning which proves this.

 

[fresh_42]

In case of integrals, it is Fubini's theorem. You also want to distinguish between infinitesimals and its limits. A sector whose area tends to zero but never is actually zero can be called an infinitesimal sector. We can add infinitely many of them and get the area of the semicircle. But it we added infinitely many times zero, it would still be zero.

Whenever operations of any kind are involved (summation, integration, differentiation, limits), you cannot automatically switch the order. What's true for finitely many doesn't necessarily hold for infinitely many.