l=lim(x tending to zero) (a-sqrt(a^2-x^2)-x^2/4)|(x^4) if l is finite then a=
i have supposed x=acostheta
$\displaystyle l= \lim_{x\to 0} \frac{a- \sqrt{a^2- x^2}- x^2/4}{x^4}$
What must a be in order that l be finite.
The kind of standard thing to do would be to "rationalize" the numerator by multiplying both numerator and denominator by $\displaystyle (a- x^2/4)+ \sqrt{a^2- x^2}$ which gives
$\displaystyle \lim_{x\to 0}\frac{(a- x^2/4)^2- (a^2- x^2)}{x^4(a- x^2/4+ \sqrt{a^2- x^2}}$
$\displaystyle = \lim{x\to 0}\frac{a^2- ax^2/2+ x^4/16- a^2+ x^2}{x^4(a- x^2/4+ \sqrt{a^2- x^2}}$
$\displaystyle = \lim_{x\to 0}\frac{(x^4)/16+ (1- a)x^2}{x^4(a- x^2/4+ \sqrt{a^2- x^2})}$
$\displaystyle = \lim_{x\to 0}\frac{1/16+ (1- a)x^{-2}}{a- x^2/4+ \sqrt{a^2- x^2}}$
Now take the limit as x goes to 0.