Question #41306

This question tests the 'Zero-Factor Cancellation Trap' combined with modulus logic.

Students frequently make the error of simplifying $\sqrt{f(x)^2}$ to $f(x)$ instead of $|f(x)|$.

In this problem, assuming $\sqrt{(\cos x - \sin x)^2} = \cos x - \sin x$ leads to a non-zero trigonometric term $2\sqrt{2}(\cos x - \sin x)$ in the numerator.

When divided by the zero-factor $\cos 2x = (\cos x - \sin x)(\cos x + \sin x)$, this incorrectly cancels out to yield a false limit component of $+2$, leading to the trap answer of $1$.

Recognizing the right-hand limit direction correctly evaluates the modulus to $\sin x - \cos x$, causing the zero-factors to perfectly annihilate before any indeterminate evaluation is needed.