Limit

Let $u = a - x$ . Then $\dfrac{\pi x}{2a} = \dfrac{\pi(a - u)}{2a} = \dfrac{\pi}{2} - \dfrac{\pi u}{2a}$ and $u \to 0$ as $x \to a$ .

The given limit can then be written as

$\displaystyle\lim_{u \to 0} u \tan\left(\dfrac{\pi}{2} - \dfrac{\pi u}{2a}\right) = \lim_{u \to 0} u \cot\left(\dfrac{\pi u}{2a}\right) = \lim_{u \to 0} \dfrac{\dfrac{2a}{\pi} \times \dfrac{\pi u}{2a}}{\tan\left(\dfrac{\pi u}{2a}\right)} = \dfrac{2a}{\pi} \lim_{y \to 0} \dfrac{y}{\tan(y)}$ ,

where $y = \dfrac{\pi u}{2a} \to 0$ as $u \to 0$ . Now $\displaystyle \lim_{y \to 0} \dfrac{\tan(y)}{y} = 1$ , so the desired limit is just $\boxed{\dfrac{2a}{\pi}}$ .