Discuss any limitation you can imagine when using the method of exaustion with construction?

Question

Please I'm stuck help me.

Answer 1

Archimedes was a master at using the method of exhaustion. They say he used it often in his derivations and proofs. He was very close to inventing the calculus, but he never crossed the bridge of going to the limit.

When you use that method, doing construction, one problem is that you can only repeat your construction -- such as bisecting a segment -- a limited number of times before the segments you're working with become too small to manipulate.

That means, that with just a few iterations, you must see the patterns emerge from what you're doing. Then you have to generalize to get a sequence or series -- perhaps infinite -- to get your answer. If you don't see the emerging pattern, or if you get it wrong, then you're stuck.

As a simple example, suppose you iterate twice and get the numbers 1 and 3. If that's all you can do, you're stuck. Starting from 1, 3 , ..., you could be looking at 1, 3, 5, 7, ... Or maybe it's 1, 3, 9, 27, 81, ... Or maybe it's 1, 3, 6, 10, 15, ...

A completely different sort of limitation is whether or not the emerging series converges or not, The series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2. But the series 1 + 1/2 + 1/3 + 1/4 + ... is divergent.

All these are considerations when you're using the method of exhaustion.

Answer 2

Construction as with compass and straightedge? Well, the lines you draw have a finite width, for one thing--so you really can't get that close to your limit. That isn't really how the method of exhaustion is used, though. It's basically a continued construction in prrinciple--the idea is that, given fine enough instruments, you COULD do it. This is enough to get you to the limit. So if this is a question your teacher is asking, it's a trick.