Lim(x*sqrt(1-cos(2*Pi/x)),x->Infinty) using analytical methods

  • Thread starter dipstik
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In summary, the speaker's cousin discussed using polygons with an increasing number of sides to derive 2*Pi*r and Pi*r^2 in geometry class. The speaker attempted to use a Taylor expansion of sine to avoid a 0 times infinity situation in finding the area, but found it to be unhelpful. They also tried L'Hopital's rule and binomial expansion, but encountered difficulties with 0*inf terms and trigonometric functions. The speaker is seeking suggestions for other methods to solve the problem.
  • #1
dipstik
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Hey guys,

My cousin came over and was talking about his geometry class and it got me to derive 2*Pi*r and Pi*r^2 using polygons of n sides as n goes to infinity. For the area I ended up using a taylor expansion of sin to weasel myself out of a 0 times infinity, but this is not being very nice. I have tried three iteration of L'Hopital and ended up with a tangent, with more trig functions to come from further iterations I think. The limit converges nicely, pretty quickly too, but I don't want to use numerical methods. I tried a binomial expansion but ended up with a long list of 0*inf. terms there.

Any ideas?
 
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  • #2
You have Cos(2*stuff) so use a double angle trig identity to get something squared in the sqrt.
 

Related to Lim(x*sqrt(1-cos(2*Pi/x)),x->Infinty) using analytical methods

1. What is the value of the limit?

The value of the limit is infinity. As x approaches infinity, the expression within the square root becomes larger and larger, resulting in an infinite value.

2. Can the limit be solved using analytical methods?

Yes, the limit can be solved using analytical methods. By applying the rules of limits and simplifying the expression, we can determine the value of the limit.

3. What is the significance of using the cosine function in the expression?

The cosine function is used in the expression to represent a periodic function. This allows us to observe the behavior of the limit as x approaches infinity, and see if it approaches a specific value or if it oscillates.

4. How can we verify the value of the limit using a graph?

We can verify the value of the limit by graphing the function and observing the behavior as x approaches infinity. The graph should show the function increasing or decreasing without bound, confirming the value of infinity.

5. Can we use substitution to solve this limit?

Yes, we can use substitution to solve this limit. By substituting a large value for x, we can see that the expression will approach infinity, confirming our answer.

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