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Math Help - Delta epsilon proof, cant solve.

  1. #1
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    Delta epsilon proof, cant solve.

    Hey, I'm having problems proving a trigonometric limit using only the delta epsilon definition. It was bundled with other relatively straightforward exercises.
    It should be solved using the definition and without using derivatives.

    limcosx= sqt2/2 as x approaches pi/4


    Can someone help?
    Last edited by HK47; April 19th 2013 at 02:57 AM.
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  2. #2
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    Re: Delta epsilon proof, cant solve.

    To prove \displaystyle \lim_{x \to \frac{\pi}{4}} \cos{(x)} = \frac{\sqrt{2}}{2}, you need to show  0 < \left| x - \frac{\pi}{4} \right| < \delta \implies \left| \cos{(x)} - \frac{\sqrt{2}}{2} \right| < \epsilon.

    To do this, we should make note of the mean value theorem, \displaystyle \frac{f(b) - f(a)}{b - a} = f'(c) for some \displaystyle c \in (a, b), and so \displaystyle \frac{\left| f(b) - f(a) \right|}{| b -a |} = \left| f'(c) \right| . That means if \displaystyle f(x) = \cos{(x)} then \displaystyle f'(x) = -\sin{(x)} and so \displaystyle f\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} and \displaystyle f'(c) = -\sin{(c)}, which is a constant value from some \displaystyle c \in \left( x, \frac{\pi}{4} \right) . Substituting gives

    \displaystyle \begin{align*} \frac{ \left| f(b) - f(a) \right|}{\left| b - a \right|} &= \left| f'(c) \right| \\ \frac{\left| \cos{(x)} - \frac{\sqrt{2}}{2} \right| }{ \left| x - \frac{\pi}{4} \right| } &= \left| -\sin{(c)} \right| \\ \left| \cos{(x)} - \frac{\sqrt{2}}{2} \right| &= \left| \sin{(c)} \right| \left| x - \frac{\pi}{4} \right| \end{align*}

    So now we can start the proof. If we let \displaystyle \delta = \frac{\epsilon}{\left| \sin{(c)} \right|} then we have

    \displaystyle \begin{align*} 0 < \left| x - \frac{ \pi}{4} \right| &< \delta \\ 0 < \left| x - \frac{\pi}{4} \right| &< \frac{\epsilon}{\left| \sin{(c)} \right| } \\ 0 < \left| \sin{(c)} \right| \left| x - \frac{\pi}{4} \right| &< \epsilon \\ 0 < \left| \cos{(x)} - \frac{\sqrt{2}}{2} \right| &< \epsilon \end{align*}

    Therefore \displaystyle 0 < \left| x - \frac{\pi}{4} \right| < \delta \implies \left| \cos{(x)} - \frac{\sqrt{2}}{2} \right| < \epsilon and thus we have proved \displaystyle \lim_{x \to \frac{\pi}{4}} \cos{(x)} = \frac{\sqrt{2}}{2}.
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    Re: Delta epsilon proof, cant solve.

    Thanks, would never have come up with the mean value theorem. I think the exercise may be misplaced in my textbook however, since it is placed with the limit definition, before the derivative is introduced. I don't think there is a way to prove this using the definition without applying the derivative.
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