Question #66383

This question tests the existence of limits for complex functions at a discontinuity when composed with a real-variable function.

The primary trap is that $\lim_{z \to \alpha+\beta} f(z)$ does not exist in the complex plane because the limit is path-dependent (it relies on the angle of approach).

Students might incorrectly conclude the limit $L$ does not exist or erroneously evaluate it along a standard path like the real axis (yielding 40).

However, because the inner function $g(x)$ restricts the domain to $x \in \mathbb{R}$, the approach to the discontinuity is locked along a specific line with direction $\alpha$.

This fixed phase shift $\frac{\alpha}{\bar{\alpha}}$ must be accounted for, rewarding students who rigorously apply the definition of the composition rather than blindly applying L'Hôpital's rule or assuming path independence.