$\lim_{x \to \pi/4} \frac{x}{4x-\pi} \int_{\pi/4}^x \frac{tan^2\theta}{\theta^2} d\theta $

Question

I need help solving this limit:

$\lim_{x \to \pi/4} \frac{x}{4x-\pi} \int_{\pi/4}^x \frac{tan^2\theta}{\theta^2} d\theta $

The limit is solvable using l'hopital's rule to get $\frac{64}{\pi^2}$, but I need to see if it is possible to do so without using it. This problem was presented to me by my friend who is taking Calculus 1 so nothing beyond simple integrals and derivatives if possible.

Answer 1

Using the mean-value theorem, we have for some $\phi \in [\pi/4,x]$

$$\begin{align} \frac{x}{4x-\pi}\int_{\pi/4}^x \frac{\tan^2(\theta)}{\theta^2}\,d\theta&=\frac{x}{4x-\pi}\left(\frac{\tan^2(\phi)}{\phi^2}\right)(x-\pi/4)\\\\ &=\frac{x}{4}\left(\frac{\tan^2(\phi^2)}{\phi^2}\right)\\\\ &\to \frac{\pi}{16}\frac{1}{(\pi/4)^2}\,\,\text{as}\,\,x\to \pi/4\\\\ &=\frac{1}{\pi} \end{align}$$

Answer 2

By the definition of derivatives and the fundamental theorem of calculus, $$ \lim_{x\to\pi/4}\frac{1}{x-\pi/4}\int_{\pi/4}^x\frac{\tan^2\theta}{\theta^2}\,dt=\frac{\tan^2(\pi/4)}{(\pi/4)^2}. $$ This means, by the product rule for limits, that your limit equals $$ \frac{\pi/4}{4}\cdot\frac{\tan^2(\pi/4)}{(\pi/4)^2}=\frac{1}{\pi}. $$

Answer 3

In a very similar spirit with the above answers, we can look at part of quotient as a definition of derivative of the function $h(x) = \displaystyle \int_{0}^x \dfrac{\tan^2 \theta}{\theta^2}d\theta$. Thus the limit you are finding is $\displaystyle \lim_{x \to \frac{\pi}{4}} \dfrac{x}{4}\left(\dfrac{h(x)-h(\frac{\pi}{4})}{x-\frac{\pi}{4}}\right)$, and this limit is the derivative at $x = \dfrac{\pi}{4}$ of $h(x) $ times $\dfrac{\pi}{16}$,and using the F.T.C as pointed out by the second answer, you have $L = \dfrac{\pi}{16}h'(\frac{\pi}{4})= \dfrac{\pi}{16}\cdot \dfrac{16}{\pi^2} = \dfrac{1}{\pi}$.