(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Limit question

1. The problem statement, all variables and given/known data

Show that the limit of [itex]f'(x)[/itex] as x --> 1 is [itex]-4/\pi[/itex]:

2. Relevant equations

[tex]f(x) = 1 - 4 \arccos\left[\frac 1 2 \left(x+\sqrt{2-x^2}\right)\right]/ \pi[/tex]

3. The attempt at a solution

[tex]f'(x)=\frac{2\sqrt 2\left(1-x/\sqrt{2-x^2}\right)}

{\left(\sqrt{1-x\sqrt{2-x^2}}\right)\pi}[/tex]

Both the numerator and the denominator --> 0 as x-->1. I tried l'Hopital's rule. The derivative of the numerator is [itex]-4\sqrt 2/(2-x^2)^{3/2}[/itex], which evaluates to [itex]-4\sqrt 2[/itex] at x = 1. To get the stated answer, the derivative of the denominator should be [itex]\pi\sqrt 2[/itex] at x=1. But it is actually

[tex]

\frac{-(1-x^2)\pi}

{\sqrt{2-x^2}\sqrt{1-x\sqrt{2-x^2}}},

[/tex]

which is [itex]0/0[/itex] at x=1. :uhh:

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# Homework Help: Limit question

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