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What is Going on Here? Would like an answer, please!

So I was trying to derive a way to approximate \(\pi\) using my compass and straightedge, and then use algebra. I began by creating a \(30\) degree angle:

So \(CAD\) is \(30\) degrees. Then, I imagined bisecting this angle into infinity. Notice if we draw a segment from \(C\) to \(D\), we get an isosceles. We can calculate for this length using the law of cosines: \(x = \sqrt{2-2\cos(\frac{30}{2r}})\), where \(r\) is a reiteration (another bisection). So, if I am correct in my assumption, \(\pi\) should be about \(x \cdot n\), if \(n\) are the number of divisions in the circle. We can find \(n\) easily enough: \(n = 24 \cdot 2^{r-2}\). we know this becuase, as we bisect, we get a table of values: \(\left \{ 24, 48, 96, \ldots \right \}\). Likewise, we find \(\theta\) thusly: \(\theta = \frac{30}{2^{r}}\). This table of values is found by dividing 360 by \(n\): \(\left \{ 30, 15, 7.5, ... \right \}\). So to find \(x\), we use the law of cosine, take the square, and multiply by n. This gives me the equation: \[\pi = \sqrt{2- 2 \cdot \cos \dfrac{30}{2^{r}}} \cdot 24 \cdot 2^{r-2}\] This makes sense from the way I constructed it, but not here: \[\lim_{r \rightarrow \infty} \sqrt{2- 2 \cdot \cos \frac{30}{2^{r}}} \cdot 24 \cdot 2^{r-2}\]

This becomes: \[\lim_{r \rightarrow \infty} \sqrt{2- 2 \cdot \cos(30 \cdot 0)} \cdot 24 \cdot 2^{r-2} = 0 \] Obviously, \(\pi \neq 0\). So, I tested with two values of \(r\) that my calculator could handle. For \(r =17\), \(\pi = 3.14159265358\) (correct to 11 decimal places.) However, at \(r = 18\), \(\pi = 3.1415\) (correct to only 4 decimal places.) So why does this equation get close to \(\pi\), as it is supposed to, and then stop, and then appear to approach zero? Thanks for the help, I am not all that great at math, and would really appreciate it!

Note by Drex Beckman
6 months, 1 week ago

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Extremely interesting! How are you saying that limit is 0? (Hint: It's an indeterminate form, (\( 0 \cdot \infty \)))
The correct limit is \( \pi \), just like you wanted. Ameya Daigavane · 6 months, 1 week ago

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@Ameya Daigavane Well, it seemed like as r approached infinity, the cosine would approach 0. of course, cos(0) = 1 and so we would get \(0 * 24 * 2^{r-2}\). Since the limit is \(\pi\), is there something wrong with my calculations? Since the precision seemed to degrade for higher r's. Thanks, I don't have the experience of ever learning limits in school, so I do not realize there is such a thing as indeterminate forms, but I was unsure what to do with the \(0 * \infty\) case. I just assumed for any number, you would get zero. Thanks for the help! :) Drex Beckman · 6 months, 1 week ago

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@Drex Beckman Oh okay, I'll try to explain then. Suppose you had two functions, \( f(x) \) and \( g(x) \),
\( \lim_{x \to \infty} f(x) = \infty \) and \( \lim_{x \to \infty} g(x) = 0 \). Now what is \( \lim_{x \to \infty} f(x)\cdot g(x)\)?
 

If you think about it, you really can't say, because it could be \( 0 \) or \( \infty \) or some value in between.
Why? It depends on the functions \( f(x) \) and \( g(x) \) themselves.
Let me give you some examples. Let \( L = \lim_{x \to \infty} f(x)\cdot g(x)\).
 

  1. \(f(x) = x\) and \(g(x) = \frac{1}{x} \Rightarrow L = 1 \) as \( f(x)g(x) = 1 \) always.
  2. \(f(x) = x^2\) and \(g(x) = \frac{1}{x} \Rightarrow L = \infty \) as \( f(x)g(x) = x \) always.
  3. \(f(x) = x\) and \(g(x) = \frac{1}{x^2} \Rightarrow L = 0 \) as \( f(x)g(x) = \frac{1}{x} \) always.
  4. \(f(x) = 5x\) and \(g(x) = \frac{1}{x} \Rightarrow L = 5 \) as \( f(x)g(x) = 5 \) always.  

So, we've seen that the limit can be anything really. This is why we call \(0 \cdot \infty \) an indeterminate form, it can 'evaluate' to anything.  

Now, to your question, how do we evaluate \(\lim_{r \rightarrow \infty} \sqrt{2- 2 \cdot \cos \frac{30}{2^{r}}} \cdot 24 \cdot 2^{r-2}\)?
Here, \( f(x) = 24 \cdot 2^{r - 2}\) and \( g(x) = \sqrt{2 - 2 \cdot \cos \frac{30}{2^r}} \) (Understand why.)
So what is L, here?
(Convert \( 30 \) to \( \frac{\pi}{6} \) radians)
L = \(\lim_{r \rightarrow \infty} \sqrt{2- 2 \cdot \cos \frac{\pi}{6 \cdot 2^{r}}} \cdot 6 \cdot 2^{r}\)
Let \( t = \dfrac{1}{6\cdot 2^r} \).
L = \( \lim_{t \rightarrow 0} \dfrac{\sqrt{2- 2 \cdot \cos \pi t }}{t} = \lim_{t \rightarrow 0} \dfrac{\sqrt{2}\sqrt{1 - \cos \pi t }}{t} = \lim_{t \rightarrow 0} \dfrac{\sqrt{2}\sqrt{2 \sin^2 \frac{\pi t}{2}}}{t} = \lim_{t \rightarrow 0} \dfrac{2 sin \frac{\pi t}{2}}{t} = \pi \)  

I used the well-known facts that \( 1 - \cos 2x = 2 \sin^2 x \) and \( \lim_{x \to 0} \frac{\sin x}{x} = 1\). Ameya Daigavane · 6 months, 1 week ago

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@Ameya Daigavane Well explained.

It's a very common misconception to "Let \( n \rightarrow \infty\) in a specific portion of the expression, while ignoring the rest of it". Calvin Lin Staff · 6 months ago

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@Ameya Daigavane Thanks a lot, I think I understand it now! Very clear explanation, also! +1 Drex Beckman · 6 months, 1 week ago

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@Drex Beckman Thanks! Also, the reason your calculator drops off in precision is because of rounding off errors and approximate values for the cosines and sines. Ameya Daigavane · 6 months, 1 week ago

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@Ameya Daigavane Right, that was what I suspected, but I put a bit too much faith in the arbitrary precision calculations, I guess. xD Drex Beckman · 6 months, 1 week ago

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