If the function f defined on ((π)/(6), (π)/(3)) by f(x) = \ array...

∵ f$\left(x\right)$ is continuous at $x = \dfrac{\pi}{4}$

$∴ f\left(\dfrac{\pi}{4}\right) = k = \lim_{x \to \dfrac{\pi}{4}}\left(\dfrac{\sqrt{2}\cos x - 1}{\cot x - 1}\right) \to \left(\dfrac{0}{0}\right)$ form

Applying L' Hospital Rule

$k = \lim_{x \to \dfrac{\pi}{4}}\left(\dfrac{\sqrt{2}\left(- \sin x\right)}{- {cosec}^{2}x}\right)$

$⇒ k = \dfrac{\sqrt{2}sin\dfrac{\pi}{4}}{{cosec}^{2}\dfrac{\pi}{4}} = \dfrac{\sqrt{2} \times \left(\dfrac{1}{\sqrt{2}}\right)}{{\left(\sqrt{2}\right)}^{2}} = \dfrac{1}{2} .$