Does L'Hospital's rule apply to complex functions?

L'Hospital's rule only applies when a limit approaches [itex]\frac{0}{0}[/itex] or [itex]\frac{\infty}{\infty}[/itex]. Niether is the case here because [itex]\lim_{x\rightarrow\infty}e^{ix}[/itex] does not exist. The real and imaginary parts oscilate between -1 and 1 as x approaches infinity. Your limit is equal to:
[tex]\lim_{x\rightarrow \infty} \frac{x-1}{e^{ipx/\hbar}} =\lim_{x\rightarrow \infty} (x-1)\cos{\frac{ipx}{\hbar}}-i(x-1)\sin{\frac{ipx}{\hbar}[/tex]
So the real and imaginary parts both oscillate between larger and larger positive and negative numbers as x gets larger, so the limit does not exist.

 

ok so the answer is infinity
we have some thing bounded in the denominator
and the numerator goes to infinity

 

I suggest you choose different boundary conditions at infinity for your QM problem.