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Thread: limits

  1. #1
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    limits and continuity

    1. Show lim Θ/sinΘ=1
    Θ->0

    (Hint: show sinΘcosΘ <Θ<tanΘ)

    I have no idea to do this problem, other than use the squeeze theorem, but i still dont know what to do. any help would be greatly appreciated.
    Last edited by johntuan; Oct 4th 2008 at 06:45 AM.
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  2. #2
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    here is a link to the classical geometric approach in determining this limit ...

    http://www.csun.edu/~ac53971/courses/math350/xtra_sine.pdf
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  3. #3
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by johntuan View Post
    1. Show lim Θ/sinΘ=1
    Θ->0

    (Hint: show sinΘcosΘ <Θ<tanΘ)

    I have no idea to do this problem, other than use the squeeze theorem, but i still dont know what to do. any help would be greatly appreciated.
    I will show my own proof for this.
    Proof.
    First draw a unit circle, and put a equilateral $\displaystyle n$-polygon in it.
    Then from the center of the circle, draw lines to the corners of the polygon.
    We see that the center angle $\displaystyle 2\pi$ is divided to $\displaystyle n$, thus all the triangles have $\displaystyle \frac{2\pi}{n}$ as the vertex of the isosceles triangle, and each triangles have the area $\displaystyle \frac{1}{2}\sin\bigg(\frac{2\pi}{n}\bigg)$.
    Hence the sum of the areas of the triangles is $\displaystyle A(n):=\frac{n}{2}\sin\bigg(\frac{2\pi}{n}\bigg)$.
    We can see that letting $\displaystyle n$ tend to infinity $\displaystyle A(n)$ tends to the are a of the circle $\displaystyle \pi$.
    That is
    $\displaystyle \lim\limits_{n\to\infty}A(n)=\pi$
    or equivalently
    $\displaystyle \lim\limits_{n\to\infty}\frac{n}{2\pi}\sin\bigg(\f rac{2\pi}{n}\bigg)=1$.
    Substitute $\displaystyle m=\frac{2\pi}{n}$, then we see that $\displaystyle m\to0$ as $\displaystyle n\to\infty$, which yields
    $\displaystyle \lim\limits_{m\to0}\frac{\sin(m)}{m}=1.$ $\displaystyle \rule{0.3cm}{0.3cm}$
    I hope this is helpful.
    Note. Here, $\displaystyle m$ travels through rational numbers and since $\displaystyle \lim$ means accumulation, we may think that $\displaystyle m$ travels through reals since it is the closure of rational numbers.
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