To show that a certain function is differentiable in $L=0$ i tried to calculate the differential quotient for $L\to 0$ $$\frac{1}{L} \cdot \left( \left(\frac{k-\sqrt{k^2+i\frac{\kappa}{L}}\frac{\left( 1-e^{i2\sqrt{k^2+i\frac{\kappa}{L}} L}\right) }{\left(1+e^{i2\sqrt{k^2+i\frac{\kappa}{L}} L} \right)}}{k+\sqrt{k^2+i\frac{\kappa}{L}}\frac{\left( 1-e^{i2\sqrt{k^2+i\frac{\kappa}{L}} L}\right) }{\left(1+e^{i2\sqrt{k^2+i\frac{\kappa}{L}} L} \right)}}\right)- \left( \frac{k-\kappa}{k+\kappa}\right)\right) $$ From Mathematica i know, that the limit exists and is given by $$\frac{2ik(-3k^2+\kappa^2)}{3(k+\kappa)^2} $$ I am trying to show this limit for days now but nothing works... i have absolutely no idea what to do and how to proceed since no calculation worked.
Is there any way to get calculation steps from Mathematica or does anyone see how one could see this limit? I would be very greatful for any help!