Goal: expected value from the moment generating function.
I've the a random variable with an exponential law, its probability density function is (parameter $\lambda=1$):
$$ f_Z(z) = e^{- z} \quad \text{if } z \geq 0 $$
$$ f_Z(z) = 0 \qquad \quad \, \text{if } z < 0 $$
The moment generating function is:
$$ \begin{align} MGF_Z(t) &= \displaystyle{\int \limits_{- \infty}^{+ \infty}} e^{t \, z} \, f_1(z) \, dz = \\ &= \displaystyle{\int \limits_{0}^{+ \infty}} e^{t \, z} \, e^{-z} \, dz = \\ &= \displaystyle{\int \limits_{0}^{+ \infty}} e^{(t - 1) \, z} \, dz = \\ &= \frac{1}{t - 1} \, \left[ e^{(t - 1) \, z} \right]_{0}^{+ \infty} \end{align} $$
$$ MGF_Z(t < -1) = \frac{1}{1 - t} $$
$$ MGF_Z(t \geq -1) = + \infty \quad $$
The expected value is:
$$ \mathbb{E}[Z] = MGF_Z'(0) $$
How can I evaluate the $MGF_Z'(0)$? The $MGF_Z(t)$ is not infinite only for $t < - 1$.