<s>[INST] Let $L=\lim_{x\to 0} \frac{ a-\sqrt {a^2-x^2} -\frac{x^2}{2}}{x^4}$, $a>0$. If $L$ is finite, find $a$ and $L$ I need a hint to start this question, because I have no idea how to do it. It’s a $\frac 00$ form, so L’Hospital can be applied, but that would be extremely tedious. Expansion can’t be used because there is no function to use it for. How do I start it?
 [/INST] You have
$$\frac{a - \sqrt{a^2-x^2}- \frac{x^2}{2}}{x^4} = \frac{a-a\sqrt{1-\frac{x^2}{a^2}} - \frac{x^2}{2}}{x^4} = \frac{a-a\left( 1- \frac{x^2}{2a^2}-\frac{x^4}{8a^4} + o(x^5)\right) - \frac{x^2}{2}}{x^4}$$
$$= \frac{\frac{x^2}{2} \left( \frac{1}{a}-1\right)+\frac{x^4}{8a^3} + o(x^5) }{x^4}$$
You see that if $a \neq 1$, the limit is infinite.
For the limite to be finite, you must have $a=1$, and then you have $$\frac{1 - \sqrt{1^2-x^2}- \frac{x^2}{2}}{x^4} = \frac{1}{8} + o(x)$$
which tends to $\frac{1}{8}$.
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