The value of $\underset{x\to a}{lim}\sqrt{a^2-x^2}\cot \left(\frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}\right) \text{is}$

$$\begin{gathered} =\underset{x\to a}{lim}\sqrt{a^2-x^2}\cot \frac{\pi }{2}\sqrt{\frac{a-x}{a+x}} \\ =\frac{2}{\pi }\cdot \underset{x\to a}{lim} (a+x)\frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}\cot \left(\frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}\right) \end{gathered}$$

$=\frac{2}{\pi }\underset{x\to a}{lim}\frac{\frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}}{\tan \frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}}(a+x)=\frac{4a}{\pi }$

$=\frac{2}{\pi }\cdot 2\mathrm{a}\left[\because \underset{\mathrm{x}\to 0}{lim} \frac{\tan \mathrm{x}}{\mathrm{x}}=1\right]$