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cot f(x) = (3x - x^3)/(1 - 3x^2) and g from (1 - x^2)/(1 + x^2): find the limit of (f(x) - f(a))/(g(x) - g(a))

The same question, asked 2 times: 16 Apr 2026 S2*, 17 Apr 2026 S2. Every wording below, oldest first.

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  1. Question 1

    16 Apr 2026 S2

    72
    If cot⁡[f(x)]=3x−x31−3x2\cot[f(x)] = \dfrac{3x - x^3}{1 - 3x^2} and sin⁡[g(x)]=1−x21+x2\sin[g(x)] = \dfrac{1 - x^2}{1 + x^2}, then lim⁡x→tf(x)−f(t)g(x)−g(t)=…\displaystyle\lim_{x \to t}\dfrac{f(x) - f(t)}{g(x) - g(t)} = \ldots

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  2. Question 2

    17 Apr 2026 S2

    73
    If f(x)=cot⁡−1(3x−x31−3x2)f(x) = \cot^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right) and g(x)=cos⁡−1(1−x21+x2)g(x) = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right), then lim⁡x→af(x)−f(a)g(x)−g(a), (0<a<12)\lim_{x \to a}\dfrac{f(x) - f(a)}{g(x) - g(a)},\ \left(0 < a < \dfrac{1}{2}\right) is

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