\begin{eqnarray*}L &=& \lim_{x\rightarrow 0}{e^{2x}-2x-1\over 4x^2}\\ &=& \lim_{x\rightarrow 0}{(e^x-1)(e^x+1)-2x\over 4x^2}\\ &=& \lim_{x\rightarrow 0}{(e^x-x-1)(e^x+1)+xe^x-x\over 4x^2\\ &=& \lim_{x\rightarrow 0}\left({e^x+1\over 4}\cdot{e^x-x-1\over x^2}+{1\over 4}\cdot{e^x-1\over x}\right)\\ &=& {1\over 2}L+{1\over 4}\end{eqnarray*}