We are tasked with evaluating the following limit:

$\lim_{x \to \frac{\pi}{4}} \frac{\cos^{20}(2x)}{20 \left( \sqrt{\cos(x)} - \sqrt{\sin(x)} \right)^{20}}$

Step 1: Analyze the behavior of the trigonometric functions as $x \to \frac{\pi}{4}$

• As $x \to \frac{\pi}{4}$, both $\cos(x)$ and $\sin(x)$ approach $\frac{1}{\sqrt{2}}$, because $\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$.

Thus, $\sqrt{\cos(x)}$ and $\sqrt{\sin(x)}$ both approach $\frac{1}{\sqrt[4]{2}}$. Therefore, the expression inside the parentheses $\sqrt{\cos(x)} - \sqrt{\sin(x)}$ tends to zero as ( x \to \frac