You have $\displaystyle P(E)= \frac{e^E+ e^{-E}}{e^E- e^{-E}}- \frac{1}{E}$ and you want to take the limit as E goes to 0 from above.
Now, what have YOU done on this? Have you tried, for example, combining the two fractions? Could you use L'Hopital's rule? Did you consider dividing both numerator and denominator by $\displaystyle e^E$?