Finding expected value from moment generating function

[jaejoon89]

Find E(X) given the moment generating function

M_X (t) = 1 / (1-t^2)

for |t| < 1.

(The pdf is f(x) = 0.5*exp(-|x|), for all x, so graphically you can see that E(X) should be 0.)

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I know that E(X) = M ' _X (t) = 0

BUT M ' _X (t) = 2x / (1-x^2)^2 which is indeterminate at 0 so maybe you need L'Hopital's rule or something but I can't get it to work. How do you do this?

 

[Mandark]

What makes you say it's indeterminate at 0?