Let f: R → R be a continuous function. Then _ x → π / 4 (π)/(4) ∫ _...

$\lim_{x \to \dfrac{\pi}{4}}\dfrac{\dfrac{\pi}{4}{∫}_{2}^{se{c}^{2}x}f\left(x\right) d x}{{x}^{2} - \dfrac{{\pi}^{2}}{16}}$

Using L-Hospital's rule, we get

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\dfrac{\pi}{4}f\left(\sec^{2}x\right) \cdot 2 \cdot \sec x \cdot \sec x \cdot \tan x - 0}{2x}$

$= \dfrac{\dfrac{\pi}{4} \cdot 2 ( \sqrt{2}{)}^{2} \cdot ( 1 ) \cdot f ( 2 )}{2 \cdot \dfrac{\pi}{4}}$

$= 2f\left(2\right)$