If lim x rightarrow 1 frac (5 x+1) 1 / 3 -(x+5) 1 / 3 (2 x+3) 1 / 2 -(x+4) 1 / 2 =frac m √(5) n(2 n) 2 / 3 , where gcd ( m , n )=1 , then 8 m +12 n is equal to

Mathematics · Limits, Continuity and Differentiability

JEE Main 2024 — 4 April, Shift 1 — Question 21

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

Answer: 100

Numerical answer — enter this value.

Step-by-step solution

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

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

$m=8, n= 3$

$8 \mathrm{~m}+12 \mathrm{n}=100$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 1
Subject: Mathematics
Chapter: Limits, Continuity and Differentiability
Topic: Indeterminate forms & its solving methods

Let f: R arrow R be a function given by f(x)=begincases1-cos 2x/x^2, & x<0,\\alpha, & x=0,\\fracbeta√(1-cos x)x, & x 0.endcases where beta…

Answer: 100

Step-by-step solution

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