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Math Help - Limit

  1. #1
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    Limit

    Why is the limit 1?
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  2. #2
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    Do you understand that \displaystyle\lim_{t\to 0}\frac{\sin(t)}{t}=1~?
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  3. #3
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    I think I see now that it is 1 when delta x is 0.0001 or some other small value near 0.
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  4. #4
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    No, it is close to 1 "when delta x is 0.0001 or some other small value near 0". The crucial point is that it keeps getting closer and closer to 1 as delta x gets closer to 0.
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  5. #5
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    You could use used L'hospital's rule to solve the limit.

    \lim \limits_{x \rightarrow 0} \dfrac{f(x)}{g(x)} = \lim \limits_{x \rightarrow 0} \dfrac{f'(x)}{g'(x)}




    \lim \limits_{\sigma x \rightarrow 0} \dfrac{\sin{\frac{\sigma x}{2}}}{\frac{\sigma x}{2}} = \lim \limits_{\sigma x \rightarrow 0} \dfrac{0.5 \cos \frac{\sigma x}{2}}{0.5}

    Substitute in \sigma x = 0 and you will get your answer.
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  6. #6
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    Whether that is valid or not depends upon exactly how you have proved that the derivative of sin(x) is cos(x). Most textbooks use the limit \displaytype\lim_{x\to 0}\frac{sin(\theta)}{\theta}= 1. If you already have that then the given limit follows immediately by letting \theta= \frac{\delta x}{2}.
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