Mathematics

Mathematics

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),0<a<12\lim_{x \rightarrow a} \frac{f(x) - f(a)}{g(x) - g(a)}, 0 < a < \frac{1}{2} is (a) 32(1+a2)\frac{3}{2(1+a^2)} (b) 32(1+x2)\frac{3}{2(1+x^2)} (c) 32\frac{3}{2} (d) −32-\frac{3}{2}
Intuitive explanation

The key is recognizing that both f(x)f(x) and g(x)g(x) have special forms hidden inside their inverse functions. The expression 3x−x31−3x2\frac{3x - x^3}{1 - 3x^2} is the tangent triple-angle formula, and 1−x21+x2\frac{1 - x^2}{1 + x^2} is the cosine double-angle formula in disguise. Once you simplify ff and gg to their underlying angles, this limit becomes a straightforward derivative ratio.

Intuitive explanation

The key is recognizing that both f(x)f(x) and g(x)g(x) have special forms hidden inside their inverse functions. The expression 3x−x31−3x2\frac{3x - x^3}{1 - 3x^2} is the tangent triple-angle formula, and 1−x21+x2\frac{1 - x^2}{1 + x^2} is the cosine double-angle formula in disguise. Once you simplify ff and gg to their underlying angles, this limit becomes a straightforward derivative ratio.

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