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(ax - (e^(kx) - 1))/(ax(e^(kx) - 1)) has a finite limit b: find a and b

The same question, asked 2 times: 2021 Paper 4*, 26 Jul 2022. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 4

    81
    If lim⁡x→0ax−(e4x−1)ax(e4x−1)\lim_{x\rightarrow 0} \frac{ax-\left( e^{4x}- 1 \right)}{ax\left( e^{4x}- 1 \right)} exists and is equal to bb, then the value of a−2ba- 2b is

  2. Question 2

    26 Jul 2022

    157
    Let β=lim⁡x→0αx−(e3x−1)αx(e3x−1)\beta =\lim_{x \rightarrow 0} \frac{\alpha x -\left( e^{3x}- 1 \right)}{\alpha x\left( e^{3x}- 1 \right)} for some α∈R\alpha \in R. Then the value of α+β\alpha + \beta is:

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