If lim x ? 1 ( 5 x + 1 ) 1 / 3 - ( x + 5 ) 1 / 3 ( 2 x + 3 ) 1 / 2 - ( x + 4 ) 1 / 2 = m 5 n ( 2 n ) 2 / 3 , where gcd ( m , n ) = 1 , then 8 m + 12 n is equal to _______ . [2024]

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Solution

Q.
If $\lim_{x \to 1} \frac{{(5 x + 1)}^{1 / 3} - {(x + 5)}^{1 / 3}}{{(2 x + 3)}^{1 / 2} - {(x + 4)}^{1 / 2}} = \frac{m \sqrt{5}}{n {(2 n)}^{2 / 3}},$ where $\gcd (m, n) = 1,$ then $8 m + 12 n$ is equal to _______ . [2024]

Ans.
(100)

Let $L = \lim_{x \to 1} \frac{{(5 x + 1)}^{1 / 3} - {(x + 5)}^{1 / 3}}{{(2 x + 3)}^{1 / 2} - {(x + 4)}^{1 / 2}} (\frac{0}{0} form)$

Using L-Hospital's rule

$L = \lim_{x \to 1} \frac{\frac{1}{3} \times 5 {(5 x + 1)}^{- 2 / 3} - \frac{1}{3} {(x + 5)}^{- 2 / 3}}{\frac{1}{2} \times 2 {(2 x + 3)}^{- 1 / 2} - \frac{1}{2} {(x + 4)}^{1 / 2}}$

$= \frac{(\frac{5}{3} - \frac{1}{3}) 6^{- 2 / 3}}{(\frac{1}{2}) 5^{- 1 / 2}} = \frac{8}{3} \times \frac{5^{1 / 2}}{6^{2 / 3}} = \frac{m \sqrt{5}}{n {(2 n)}^{2 / 3}}$

$\Rightarrow m = 8, n = 3 \Rightarrow 8 m + 12 n = 100$

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