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

Homework Helper
Gold Member

## Main Question or Discussion Point

I have to evaluate

$$\lim_{x\rightarrow \infty} \frac{x-1}{e^{ipx/\hbar}}$$

is this equal to

$$\lim_{x\rightarrow \infty} \frac{\hbar}{ip e^{ipx/\hbar}}$$

??

LeonhardEuler
Gold Member
L'Hospital's rule only applies when a limit approaches $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Niether is the case here because $\lim_{x\rightarrow\infty}e^{ix}$ does not exist. The real and imaginary parts oscilate between -1 and 1 as x approaches infinity. Your limit is equal to:
$$\lim_{x\rightarrow \infty} \frac{x-1}{e^{ipx/\hbar}} =\lim_{x\rightarrow \infty} (x-1)\cos{\frac{ipx}{\hbar}}-i(x-1)\sin{\frac{ipx}{\hbar}$$
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.

Homework Helper
Gold Member
Dang. I guess I was trying a little too hard to make this QM problem work :tongue2:

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.