If _ x arrow 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______

Correct answer: 0 aligned & _ x arrow 1 (1)/(3)(5 x+1)^ -2 / 3 5-(1)/(3)(x+5)^ -2 / 3 (1)/(2)(2 x+3)^ -1 / 2 · 2-(1)/(2)(x+4)^ -1 / 2 & =(8)/(3) 5 6^ 2 / 3 array l m=8 n=3 array & 8 m+12 n=100 aligned

JEE Main 20244 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 4 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Limits

JEE Main 2024 · 4 April · Morning Shift · Mathematics · Limits

If _ x arrow 1 (5 x+1)^ 1 / 3 -(x+5)^ 1 / 3 (2 x+3)^ 1 / 2 -(x+4)^...

\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}

, where

\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1

, then

8 \mathrm{~m}+12 \mathrm{n}

is equal to______

Correct answer

Step-by-step solution

\begin{aligned} & \lim _{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}} \begin{array}{l}m=8 \\ n=3\end{array} \\ & 8 m+12 n=100\end{aligned}

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