# Math Help - Delta epsilon proof, cant solve.

1. ## 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?

2. ## 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 $\displaystyle \lim_{x \to \frac{\pi}{4}} \cos{(x)} = \frac{\sqrt{2}}{2}$.

3. ## 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.