Let $L=\underset{x\to 0}{\lim }\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4},a>0$ if $L$ is finite, then

$$\begin{gathered} L=\underset{x\to 0}{\lim }\frac{a-a{(1-\frac{x^2}{a^2})}^{\frac{1}{2}}-\frac{x^2}{4}}{x^4} \\ =\underset{x\to 0}{\lim }\frac{a-a(1-\frac{1}{2}\cdot \frac{x^2}{a^2}+\frac{\frac{1}{2}(\frac{1}{2}-1)}{2!}\cdot \frac{x^4}{a^4}\cdots )-\frac{x^2}{4}}{x^4} \\ =\underset{x\to 0}{\lim }\frac{a-a+\frac{a{x}^2}{2a^2}+\frac{x^4}{8a^3}\cdots -\frac{x^2}{4}}{x^4} \\ \text{It will be finite if coefficient of }{x}^2=0 \\ \Rightarrow \frac{1}{2a}-\frac{1}{4}=0 \\ \Rightarrow \frac{1}{2a}=\frac{1}{4} \\ \therefore a=2 \\ L=\frac{1}{8a^3}=\frac{1}{8\left(2^3\right)}=\frac{1}{64} \end{gathered}$$