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Math Help - limit

  1. #1
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    limit

    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
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  2. #2
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    Re: limit

    Could you write the question a little clearer?
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  3. #3
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    Re: limit

    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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