If lim x→1 (5x+1)^1/3 − (x+5)^1/3 / (2x+3)^1/2 − (x+4)^1/2 = m√5 / n(2n)^{2/3}, where gcd(m, n) = 1,

Question

If \(\lim\limits _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(8 \mathrm{m}+12 \mathrm{n}\) is equal to _____.

Answer 1

Correct answer: 100

\(\lim \limits_{x \rightarrow 1} \frac{\frac{1}{3}(5 x+1)^{-2 / 3} 5-\frac{1}{3}(x+5)^{-2 / 3}}{\frac{1}{2}(2 x+3)^{-1 / 2} \cdot 2-\frac{1}{2}(x+4)^{-1 / 2}}\)

\(=\frac{8}{3} \frac{\sqrt{5}}{6^{2 / 3}}\)

\( \mathrm{m}=8\)

\(\mathrm n = 3\)

\(8 m+12 n=100\)