Let the slope of the line 45 x+5 y+3=0 be 27 r₁+(9 r₂)/2 for some r₁, r₂ ∈ ℝ. Then lim x → 3(∫₃ˣ (8 t²)/((3 r₂ x)/2-r₂ x²-r₁ x³-3 x) d t) is equal to _________.

Solution

Maths sir · 5 steps

The bracket only depends on x, so pull it out of the integral and you have a clean 0/0 limit for L'Hopital.

1. Use the slope. The line 45x + 5y + 3 = 0 has slope -45/5 = -9. So 27r₁ + 9r₂/2 = -9. You do not need r₁ and r₂ separately, only this combination.
2. Pull out the denominator. The denominator D(x) = 3r₂x/2 - r₂x² - r₁x³ - 3x has no t in it, so it comes out of the integral. The expression becomes N(x)/D(x) with N(x) = ∫₃ˣ 8t² dt.
3. Check the form. N(3) = 0. Also D(3) = 9r₂/2 - 9r₂ - 27r₁ - 9 = -(27r₁ + 9r₂/2) - 9 = 9 - 9 = 0. This is the 0/0 form, so L'Hopital's rule applies.

   lim(x → 3) N(x)/D(x) = N'(3)/D'(3) = 8x²/(3r₂/2 - 2r₂x - 3r₁x² - 3)|ₓ₌₃

4. Differentiate both. By the fundamental theorem of calculus, N'(3) = 8 × 9 = 72. For the denominator, D'(3) = 3r₂/2 - 6r₂ - 27r₁ - 3 = -(27r₁ + 9r₂/2) - 3 = 9 - 3 = 6.
5. So. The limit is 72/6 = 12.

The common slip: 8. Dropping the -3 when differentiating -3x makes D'(3) = 9 and gives 72/9 = 8, not 12.

The idea to keep. A variable-limit integral over a function that vanishes at the same point is a 0/0 limit. Differentiate the integral with the fundamental theorem, and keep any given relation as one combined block.

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