Consider the Moment Generating Function of the Uniform distribution $U(a,b)$ given by $$M(t)=\frac{e^{tb}-e^{ta}}{t(b-a)} \mbox{if }t\ne 0; M(0)=1.$$
Now, it is well known that if MGF is finite in an interval around $0$, then derivatives of all orders exists (see for instance, Billingsley's book Probability and Measure pg 278). But here $M'(0)=\lim_{t\to 0}\frac{M(t)-1}{t}$ does not seem to exist.
Why is that?
$$\frac{M(t)-1}{t}=\frac{1}{b-a}\frac{e^{tb}-e^{ta}-t(b-a)}{t^2}$$ L'Hospital rule gives $$\frac{M(t)-1}{t}\sim\frac{1}{b-a}\frac{be^{tb}-ae^{at}-(b-a)}{2t}\sim\frac{1}{b-a}\frac{b^2e^{bt}-a^2e^{at}}{2}\xrightarrow{t\rightarrow0}\frac{b+a}{2}$$