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limit of (sqrt(x^2 - x + 1) - ax) at infinity equals b: find a and b

The same question, asked 2 times: 2021 Paper 21*, 25 Jul 2022. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 21

    80
    If lim⁡x→∞(x2−x+1−ax)=b\lim_{x\rightarrow \infty} \left( \sqrt{x^{2}-x+ 1}-ax \right)=b, then the ordered pair (a,b)(a,b) is:

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  2. Question 2

    25 Jul 2022

    65
    If lim⁡n→∞(n2−n−1+nα+β)=0\lim_{n \rightarrow \infty} \left( \sqrt{n^{2}- n - 1}+ n\alpha + \beta \right)= 0 then 8(α+β)8(\alpha + \beta) is equal to:

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