I am having problems with an integral while trying to find the mgf of the exponential pdf
$ \int_0^{\infty} e^{tx} \beta ^{-1}e^{-x/ \beta}$ where $\beta$ and $t$ are constants.
I can get to this stage:
$\beta^{-1} \frac{1}{t -1/ \beta} e^{(t-(1/\beta))x}$ evaluated from 0 to $\infty$, however I do not understand why it does not evaluate to infinity (assuming my integration was correct). Rather the book gives $ \frac{1}{1 - t \beta}$.
Help would be much appreciated.
$$\int_0^\infty e^{\left(t-\frac1\beta\right)x}\beta^{-1}dx=\left.\frac1{\beta t-1} e^{\left(t-\frac1\beta\right)x}\right|_0^\infty=\begin{cases}\infty,&t-\frac1\beta\ge0\\{}\\\frac1{\beta t-1},&t-\frac1\beta<0\end{cases}$$
since for example
$$t-\frac1\beta>0\implies e^{\left(t-\frac1\beta\right)x} \xrightarrow[x\to\infty]{}\infty$$
You try the other cases.