A beautiful limit

$\begin{aligned} S & = \lim_{x \to 0} \frac {a-\sqrt{a^2-x^2}-\frac {x^2}4}{x^4} \\ & = \lim_{x \to 0} \frac {a-a{\color{#3D99F6}\sqrt{1-\frac {x^2}{a^2}}}-\frac {x^2}4}{x^4} & \small \color{#3D99F6} \text{By Maclaurin series} \\ & = \lim_{x \to 0} \frac {a-a{\color{#3D99F6}1-\frac {x^2}{2a^2} - \frac {x^4}{8a^4} - O(x^6)}-\frac {x^2}4}{x^4} & \small \color{#3D99F6} \text{Put }a=2 \\ & = \lim_{x \to 0} \frac {2-2+\frac {x^2}4 + \frac {x^4}{64} + O(x^6)-\frac {x^2}4}{x^4} \\ & = \lim_{x \to 0} \frac {\frac {x^4}{64} + O(x^6)}{x^4} & \small \color{#3D99F6} \text{Divide up and down by }x^4 \\ & = \lim_{x \to 0} \frac {\frac 1{64} + O(x^2)}1 \\ & = \frac 1{64} \end{aligned}$

$\implies aL = \frac 2{64} = 0.03125 \approx \boxed{0.03}$