Limits, Continuity & Differentiability
L'Hôpital Rule — Limit with Integral and Polynomial Denominator
nta_pyq_2024_jan
Grade 12
Question:
Let the slope of the line $45x+5y+3=0$ be $27r_1+\dfrac{9r_2}{2}$ for some $r_1,r_2\in\mathbb{R}$. Then $\displaystyle\lim_{x\to3}\left(\int_3^x\dfrac{8t^2}{\frac{3r_2x}{2}-r_2x^2-r_1x^3-3x}\,dt\right)$ is equal to
Step-by-Step Solution
Key Concept: The slope of $45x+5y+3=0$ is $-9$, so $27r_1+9r_2/2=-9$. Apply L'Hôpital's rule as $x\to3$: numerator becomes $8x^2$ at $x=3$, denominator becomes its derivative at $x=3$.
Slope $=-9$, so $27r_1+9r_2/2=-9$. L'Hôpital gives $\frac{72}{-9r_2/2-27r_1-3}=\frac{72}{9-3}=12$.
Correct Answer: 12