Why $m(t)=\frac{t^2}{1-t^2},t<1$,cannot be a moment-generating function?

Question

$m(t)=\frac{t^2}{1-t^2},t<1$

The suggested answer provided by our teacher states that m(0)=0, so there's no real value with this function as mgf; hence mgf doesn't exist.

Intuitively I understand that $e^{xt}>0$ should always be true, so the expectation should always be positive under this transformation. But I still don't quite get what does "m(0)=0 so there's no real value to this function" and how this statement backed the conclusion "mgf does not exist".

Answer 1

For any mgf $M$, $M(0)=1$. Since $m(0)=0\ne 1=M(0)$, $m$ is not an mgf.

Answer 2

$$m_X(t)=\mathbb Ee^{tX}\text{ so that }m_X(0)=\mathbb Ee^{0\cdot X}=\mathbb E1=1$$

showing that for every moment generating function we have $m(0)=1\neq0$.