Characteristic function of an exponential distribution

[Zaare]

I need to calculate the characteristic function of an exponential distribution:
$$\phi _X \left( t \right) = \int\limits_{ - \infty }^\infty {e^{itX} \lambda e^{ - \lambda x} dx} = \int\limits_{ - \infty }^\infty {\lambda e^{\left( {it - \lambda } \right)x} dx}$$

I have arrived at the following expression:
$$\frac{{i\lambda }}{{i\lambda + t}}\mathop {\lim }\limits_{x \to \infty } \left( {e^{\left( {\lambda - it} \right)x} } \right)$$

and I can't calculate the limit.
Any help would be appreciated.

 

[Zaare]

Nevermind this one, I had overlooked something in my calculations. I've solved it.