Limits, Continuity & Differentiability
L'Hôpital / Asymptotic Expansion near Root of Quadratic
nta_pyq_2023_apr
Grade 12
Question:
If $\alpha>\beta>0$ are roots of $ax^2+bx+1=0$, and $\displaystyle\lim_{x\to 1/\alpha}\left(\dfrac{1-\cos(x^2+bx+a)}{2(1-\alpha x)^2}\right)^{1/2}=\dfrac{1}{k}\left(\dfrac{1}{\beta}-\dfrac{1}{\alpha}\right)$, then $k$ is equal to
$2\beta$
$\alpha$
$2\alpha$
$\beta$
Step-by-Step Solution
Key Concept: Roots of $x^2+bx+a=0$ are $\frac{1}{\alpha},\frac{1}{\beta}$. Near $x=\frac{1}{\alpha}$: $x^2+bx+a\approx(x-\frac{1}{\alpha})(x-\frac{1}{\beta})$. Use $1-\cos u\approx\frac{u^2}{2}$ as $u\to0$.
$k=2\alpha$.
Correct Answer: 3