What would this quantity be?

[tex] \lim_{t \rightarrow \infty} e^{-i \alpha |x - t|} \cdot (|x -t| - 1) - \lim_{t \rightarrow - \infty} e^{-i \alpha |x - t|} \cdot (|x -t| - 1) = ? [/tex]

It looks to me like it is just zero, but I was hoping it would be:

[tex] \frac{2e^{-i \alpha x}}{1 + \alpha^2} [/tex]

where [tex] \alpha [/tex] is a real number, since this was the last step in proving that

[tex] f(t) = e^{-i \alpha t} [/tex]

is an eigenfunction of the kernel:

[tex] K(x,t) = e^{-i \alpha |x - t|} [/tex]

with an eigenvalue:

[tex] \lambda = \frac{2}{1 + \alpha^2} [/tex]

Perhaps I solved my integral wrong or made a mistake somewhere.