Characteristic function of $X\sim \text{exp}(\lambda)$

Question

If $X\sim \text{exp}(\lambda)$ then the characteristic function is $$\phi_X(t) =\frac{1}{\lambda}\int_0^\infty e^{idx}e^{-x/\lambda} = \frac{1}{\lambda i t - 1}e^{(it - \frac{1}{\lambda})x}\Big\vert_0^\infty =\frac{1}{1-i\lambda t}$$ Where $\frac{1}{1-i\lambda t}$ can be separated into an imaginary and complex part.

I don't understand how $$\frac{1}{\lambda i t - 1}e^{(it - \frac{1}{\lambda})x}\Big\vert_0^\infty =\frac{1}{1-i\lambda t}$$ Specifically, why don't we have to worry about $e^{(it-1/\lambda)\cdot\infty}$?

Thanks.

Answer 1

You do have to worry about it. Compute the limit on the $x \to \infty$ side; what do you get?