Let's say I have a probability density function of $f_X(x)=\frac{x}{3}, \; \forall x\in(0,3)$ for some continuous random variable $X$.
Now I wish to find its moment generating function, I know that for something to satisfy the conditions of a mgf it must have some finite answer for $\int_{-\infty}^{\infty}e^{tx}f_X(x)dx$. My issue is that when I evaluate this integral, it diverges. Should my bounds for integration be my bounds for $x$, or have I made some error in understanding?