Binomial expansion

If $1+x^4+x^5=\displaystyle\sum_{i=0}^5a_i(1+x)^i$, for all $x$ in $\mathbb R$, then $a_2$ is :

Let
$$L=\lim_{x\to0}\frac{a-\sqrt{a^2-x^2}-\dfrac{x^2}4}{x^4},\qquad a>0.$$
If $L$ is finite, then