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

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SUMMARY

The discussion focuses on evaluating the limit Lim(x*sqrt(1-cos(2*Pi/x)),x->Infinity) using analytical methods. The user initially attempted to derive the limit using Taylor expansion and L'Hopital's Rule but encountered complications with indeterminate forms. The suggestion to apply double angle trigonometric identities was proposed as a potential solution to simplify the expression. The limit converges quickly, indicating that analytical methods can yield results without resorting to numerical techniques.

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  • Understanding of limits in calculus
  • Familiarity with Taylor series expansions
  • Knowledge of L'Hopital's Rule
  • Proficiency in trigonometric identities, particularly double angle formulas
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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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You have Cos(2*stuff) so use a double angle trig identity to get something squared in the sqrt.
 

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