# Does this "theorem" of limits hold in general?

• I
• walking
In summary, the conversation discusses the process of finding the centre of mass of a semicircle using polar coordinates and the mistake made by assuming the sectors could be modeled as straight lines with 0 area. The important lesson learned is that when dealing with limits, one should not take limits for one thing before taking limits for another thing. This is known as Fubini's theorem in the case of integrals. Additionally, it is important to distinguish between infinitesimals and their limits, as adding infinitely many of them can result in a non-zero value, while adding infinitely many zeros will still result in zero. The reasoning behind this is that operations involving infinitely many terms may not follow the same rules as operations involving finitely many terms.
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.

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.

## 1. What is a theorem of limits?

A theorem of limits is a mathematical statement that describes the behavior of a function as its input approaches a certain value. It helps us understand the behavior of a function near a specific point.

## 2. How do you determine if a theorem of limits holds in general?

To determine if a theorem of limits holds in general, we must prove that it holds for all possible values of the function's input. This can be done using mathematical techniques such as proof by contradiction or proof by induction.

## 3. Are there any exceptions to the theorem of limits?

Yes, there are some functions that do not follow the theorem of limits. These are typically functions that have discontinuities or behave in a non-standard way near the point of interest.

## 4. Can the theorem of limits be applied to all types of functions?

Yes, the theorem of limits can be applied to all types of functions, including polynomial, exponential, trigonometric, and logarithmic functions. However, the techniques used to prove the theorem may differ depending on the type of function.

## 5. Why is the theorem of limits important in mathematics?

The theorem of limits is important because it allows us to make precise statements about the behavior of functions near a specific point. This is useful in many areas of mathematics, such as calculus, where we need to understand how functions change and approach certain values.

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