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Limit of a difference of fractional powers (roots)

This type of question, asked 3 times: 2021 Paper 17, 27 Jul 2022, 4 Apr 2024. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    71
    The value of lim⁡x→0(x1−sin⁡x8−1+sin⁡x8)\lim_{x\rightarrow 0} \left( \frac{x}{\sqrt[8]{1 - \sin x}-\sqrt[8]{1 + \sin x}} \right) is equal to:

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  2. Question 2

    27 Jul 2022

    155
    If for p≠q≠0p \neq q \neq 0, then function f(x)=p(729+x)7−3729+qx3−9f(x) =\frac{\sqrt[7]{p(729 + x)}- 3}{\sqrt[3]{729 + qx}- 9} is continuous at x=0x = 0, then:

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  3. Question 3

    4 Apr 2024

    81
    If lim⁡x→1(5x+1)1/3−(x+5)1/3(2x+3)1/2−(x+4)1/2=m5n(2n)2/3\lim_{x\rightarrow 1} \frac{(5x+ 1)^{1/3}- (x+ 5)^{1/3}}{(2x+ 3)^{1/2}- (x+ 4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}, where gcd(m,n)=1gcd(m,n) = 1, then 8 m+12n8\text{ }m + 12n is equal to