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 _______.

Solution

Maths sir · 4 steps

Both top and bottom vanish at x = 1, so differentiate each once and then match the answer's shape.

1. Check the form. At x = 1 the numerator is 6^(1/3) - 6^(1/3) = 0 and the denominator is √5 - √5 = 0. It is a 0/0 form, so L'Hopital's rule applies.
2. Differentiate both. Numerator: 5/3(5x+1)^(-2/3) - 1/3(x+5)^(-2/3), which at x = 1 is 4/3 · 6^(-2/3). Denominator: (2x+3)^(-1/2) - 1/2(x+4)^(-1/2), which at x = 1 is 1/2√5.

   lim(x → 1) = (4/3 · 6^(-2/3))/(1/2√5) = 8√5/(3 · 6^(2/3))

3. Match the form. Compare with m√5/n(2n)^(2/3). Taking n = 3 makes 2n = 6, so m = 8, and gcd(8, 3) = 1 as required.
4. So. 8m + 12n = 64 + 36 = 100.

The idea to keep. For a 0/0 limit with roots, differentiate top and bottom separately and do not forget the chain-rule factor from inside each bracket.

Answer from the NTA final answer key. Solution by PrepWiser.