limit

**prasum** · June 24th 2011, 12:19 AM

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

**Glitch** · June 24th 2011, 03:33 AM

Could you write the question a little clearer?

**HallsofIvy** · June 24th 2011, 10:16 AM

$l= \lim_{x\to 0} \frac{a- \sqrt{a^2- x^2}- x^2/4}{x^4}$
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 $(a- x^2/4)+ \sqrt{a^2- x^2}$ which gives
$\lim_{x\to 0}\frac{(a- x^2/4)^2- (a^2- x^2)}{x^4(a- x^2/4+ \sqrt{a^2- x^2}}$

$= \lim{x\to 0}\frac{a^2- ax^2/2+ x^4/16- a^2+ x^2}{x^4(a- x^2/4+ \sqrt{a^2- x^2}}$

$= \lim_{x\to 0}\frac{(x^4)/16+ (1- a)x^2}{x^4(a- x^2/4+ \sqrt{a^2- x^2})}$

$= \lim_{x\to 0}\frac{1/16+ (1- a)x^{-2}}{a- x^2/4+ \sqrt{a^2- x^2}}$

Now take the limit as x goes to 0.

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