Question #41300

This question constructs a 'Lethal Collision' between the domain boundaries of inverse trigonometric functions and the behavior of transcendental functions near zero.

Students often blindly apply Taylor series expansions (e.g., $e^{x^2} - x^2 \approx 1 + x^4/2$) and assume that since the argument approaches 1, the limit evaluates to 0.

The trap lies in ignoring the strict inequality $e^{x^2} - x^2 > 1$ for all $x \neq 0$.

Because the argument strictly exceeds the principal value branch domain $[-1, 1]$ of $\cos^{-1}$, the function is entirely undefined in any punctured neighborhood of zero, making the limit non-existent.

The inclusion of the Greatest Integer Function $[x^2]$ serves as secondary DNA to obfuscate the argument and force piecewise consideration, further distracting from the domain violation.