QJEEM24Ma3829: JEE Main 2024 Mathematics - Class 12 JEEM24Ma3829 Solution: If lim x → 1 ( 5 x + 1 ) 1 / 3 − ( x + 5 ) 1 / 3 ( 2 x + 3 ) 1…

$$I=\underset{x\to 1}{lim}\frac{(5x+1{)}^{1/3}-(+5{)}^{1/}}{(2x+3{)}^{1/2}-(x+4{)}^{1/2}}$$

From: $\frac{0}{0}$ , using $\mathrm{L}-\mathrm{H}$ rule

$\begin{aligned}& I=\lim _{x \rightarrow 1} \frac{\frac{1}{3} \times 5(5 x+1)^{-2 / 3}--(+5)}{\frac{1}{2} \times 2(2 x+3)^{-1 / 2}-\frac{1}{2}(x+4)^{-1 / 2}} \\& =\frac{\left(\left.\frac{5}{3}-\frac{1}{3} \right\rvert\, 6^{-2 / 3}\right.}{\frac{1}{2} 5^{-1 / 2}}=\frac{8}{3} \times \frac{5^{1 / 2}}{6^{2 / 3}}=\frac{m \sqrt{5}}{n(2 n)^{2 / 3}} \\& \Rightarrow m=8, n=3 \\& \Rightarrow 8 m+12 n=100\end{aligned}$