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Math Help - Limit and area of a circle

  1. #1
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    Limit and area of a circle

    I have the following expression:


    How do you show that:

    Thanks!
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  2. #2
    MHF Contributor
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    If you rewrite it a little, it becomes familiar.

    \frac{1}{2} \cdot n \cdot r^{2} \cdot \sin\left(\frac{2 \cdot \pi}{n}\right)


    \frac{1}{2} \cdot r^{2} \cdot \frac{\sin\left(\frac{2 \cdot \pi}{n}\right)}{\frac{1}{n}}


    r^{2} \cdot \frac{\sin\left(\frac{2 \cdot \pi}{n}\right)}{\frac{2}{n}}


    \pi \cdot r^{2} \cdot \frac{\sin\left(\frac{2 \cdot \pi}{n}\right)}{\frac{2 \cdot \pi}{n}}
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  3. #3
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    Quote Originally Posted by ty2391 View Post
    I have the following expression:


    How do you show that:

    Thanks!

    For the following proof you need to know that \lim_{x\to 0}\frac{sin x}{x}=1, and using continuity or L'Hospital's rule, we can prove then that in fact \lim_{h(n)\to 0}\frac{sinh(n)}{h(n)} if h(n) is a sequence converging to zero, so:

    \lim_{n\to 0}\frac{1}{2}nr^2sin\frac{2\pi}{n}=\lim_{n\to 0}\frac{2\pi r^2}{2}\,\frac{sin\frac{2\pi}{n}}{\frac{2\pi}{n}}=  \pi r^2\cdot 1=\pi r^2

    Tonio
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