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Limit (x^n f(a) - a^n f(x))/(x - a) using f(a) and f'(a)

This type of question, asked 3 times: 2021 Paper 16, 2021 Paper 24, 2021 Paper 26. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 16

    42
    lim⁡x→2x2f(2)−4f(x)(x−2)\lim_{x\rightarrow 2} \frac{x^{2}f(2) - 4f(x)}{(x- 2)}, where f(2)=4f(2) = 4 and f′(2)=1f^{'}(2) = 1, is equal to

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  2. Question 2

    2021 Paper 24

    82
    Let f(x)=x6+2x4+x3+2x+3,x∈Rf(x) =x^{6}+ 2x^{4}+x^{3}+ 2x+ 3,x\in R. Then the natural number nn for which lim⁡x→1xnf(1)−f(x)x−1=44\lim_{x\rightarrow 1} \frac{x^{n}f(1) -f(x)}{x- 1}= 44 is

  3. Question 3

    2021 Paper 26

    63
    Let f(x)f(x) be a differentiable function at x=ax = a with f′(a)=2f^{'}(a) = 2 and f(a)=4f(a) = 4. Then lim⁡x→axf(a)−af(x)x−a\lim_{x \rightarrow a} \frac{xf(a) - af(x)}{x - a} equals:

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