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$$\lim_{\omega \to \infty } \int_0^{\delta} \left| \frac{\cos \left(\frac{t}{2}\right)-\cos (\omega t)}{t}\right| dt$$

$$\delta<2\pi$$ I know it is infinity, but would like to know the growth of it as $\omega \to \infty$

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    @DavidP : you mean substitute $\omega t = \alpha$? – 2017-01-07
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    After graphing out the integrand for a few values of $\omega$, it looks like most of the value is coming from close to $t=2\pi$, so perhaps you should examine that more closely. – 2017-01-07
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    Upon looking at a few more values of $\omega$, I'm not so sure that limit exists. When $\omega$ is an integer, $\cos(\omega t) \to 1$ as $t \to 2\pi$ so the integral is large, but for $\omega$ away from an integer, that no longer holds true and the integral is smaller and more dominated by its behavior away from $t=2\pi$, so the integral is fluctuating. Why do you believe the limit is $\infty$? – 2017-01-07
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    $O(\log(\omega))$ – 2017-01-07
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    @CodeLabMaster : I don't want the upper limit of the definite integral to be $2\pi$, a $\delta < 2\pi$ would do. – 2017-01-07
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    @CodeLab : the denominator should be $t$ as this is a continous version. In descrete case it is always intezer and limit exists as you have mentioned but not othervalues. Now this problem is eliminated i hope. Also this is the correct version in continuos case. – 2017-01-07

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Based on your comment, you already have the estimate that it grows $O(\log \omega)$, but here's a formal derivation of it in case it's helpful.

As long as we're close enough to $0$, then we can use $\cos x \sim 1 - \frac{x^2}{2}$, but since we have a term $\cos \omega t$ and $\omega \to \infty$, $t$ has to keep getting closer to $0$ to make the approximation accurate. Let's choose $\epsilon > 0$ such that $\omega t < \epsilon$. Then $t < \epsilon / \omega$ and we can break our integral into

$$\color{red}{\int_0^{\epsilon / \omega} \left| \frac{ \left[ 1 - \left( \frac{t}{2} \right)^2 \right] - \left[ 1 - \left( \omega t \right)^2 \right]}{t} \right| \, dt} + \color{blue}{\int_{\epsilon / \omega}^\delta \left| \frac{\cos \left( \frac{t}{2} \right) - \cos(\omega t)}{t} \right| \, dt}$$

Simplifying the first integral, we find (as long as $\omega \geq 1/2$)

$$\color{red}{\int_0^{\epsilon / \omega} \left| \frac{ \left( \omega t \right)^2 - \left( \frac{t}{2} \right)^2}{t} \right| \, dt = \int_0^{\epsilon / \omega} \left( \omega^2 - \frac{1}{4} \right) t \, dt = \left( 1 - \frac{1}{4 \omega^2} \right) \frac{\epsilon^2}{2}}$$

Now that we are avoiding the sigularity in the second integral, we can bound the numerator and directly evaluate the result

$$\color{blue}{\int_{\epsilon / \omega}^\delta \left| \frac{\cos \left( \frac{t}{2} \right) - \cos(\omega t)}{t} \right| \, dt \leq 2 \int_{\epsilon / \omega}^\delta \frac{1}{t} \, dt = 2 \left( \log \delta - \log \epsilon + \log \omega \right) }$$

Putting everything back together gives us $O( \log \omega )$.