If and , then , where ,…

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

If f(x)=cot⁡−1(3x−x31−3x2)f(x)=\cot^{-1}\left(\frac{3x-x^3}{1-3x^2}\right) and g(x)=cos⁡−1(1−x21+x2)g(x)=\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), then lim⁡x→af(x)−f(a)g(x)−g(a)\displaystyle\lim_{x\to a}\frac{f(x)-f(a)}{g(x)-g(a)}, where 0<a<120<a<\frac12, is (a) 32(1+a2)\frac{3}{2(1+a^2)} (b) 32(1+x2)\frac{3}{2(1+x^2)} (c) 32\frac32 (d) −32-\frac32

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