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Let us consider 

MultiSeries:-series(Psi((2*x+1)/(2*x))-Psi((x+1)/(2*x)), x = 0);

x-(1/2)*x^2+(1/4)*x^4-(1/2)*x^6 +O(x^7)

The above result contradicts 

MultiSeries:-limit(diff(Psi((2*x+1)/(2*x))-Psi((x+1)/(2*x)), x), x = 0);
                           undefined
MultiSeries:-limit((Psi((2*x+1)/(2*x))-Psi((x+1)/(2*x)))/x, x = 0, right);
                               1
MultiSeries:-limit((Psi((2*x+1)/(2*x))-Psi((x+1)/(2*x)))/x, x = 0, left);
                           undefined
plot((Psi((2*x+1)/(2*x))-Psi((x+1)/(2*x)))/x, x = -0.1e-1 .. 0.1e-2, discont, y = -5 .. 5);

multiseries

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