L=$\underset{x\to 0}{Lt}\frac{a-\sqrt{a^2-x^2}-{\displaystyle \frac{x^2}{4}}}{x^4}, a>0.$ If L is finite then

$\begin{array}{l}L=\underset{x\to 0}{Lt}\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}isfinite\\ =\underset{x\to 0}{Lt}\frac{\frac{\left(a-\sqrt{a^2-x^2}\right)\left(a+\sqrt{a^2-x^2}\right)}{a+\sqrt{a^2-x^2}}}{x^4}\\ \underset{x\to 0}{Lt}\frac{\frac{a^2-a^2+x^2}{a+\sqrt{a^2-x^2}}-\frac{x^2}{4}}{x^4}\\ =\underset{x\to 0}{Lt}\frac{1}{x^4}\left(\frac{x^2}{a+\sqrt{a^2-x^2}}-\frac{x^2}{4}\right)\\ =\underset{x\to 0}{Lt}\frac{1}{x^2}\left(\frac{1}{a+\sqrt{a^2-x^2}}-\frac{1}{4}\right)\\ =\underset{x\to 0}{Lt}\frac{1}{x^2}\left(\frac{4-a-\sqrt{a^2-x^2}}{4\left(a+\sqrt{a^2-x^2}\right)}\right)\\ Dr\to 0Nr\to 0\\ 4-a-a=0\\ 2a=4\Rightarrow a=2\end{array}$