# Moment Generating Function of an Exponential variable

I know that when if we have an exponential random variable with parameter $$\lambda$$, the moment generating function is $$\frac{\lambda}{\lambda-t}$$ when $$t < \lambda$$, but what can I say about the function when $$t \ge \lambda$$? Based on my computation of the integral, I think it is $$+\infty$$, but I'm not too sure about this. Thank you

It is $$\infty$$ becasue $$\lambda \int_0^{\infty} e^{(t-\lambda)x} dx \geq \lambda \int_0^{\infty} 1 dx=\infty$$.