Limits, Continuity & Differentiability
Limit of Integral-Defined Function — L'Hôpital
nta_pyq_2024_apr
Grade 12
Question:
Let $f(x)=\displaystyle\int_0^x(t+\sin(1-e^t))\,dt$, $x\in\mathbb{R}$. Then $\displaystyle\lim_{x\to0}\frac{f(x)}{x^3}$ is equal to
$-\dfrac{1}{6}$
$\dfrac{2}{3}$
$-\dfrac{2}{3}$
$\dfrac{1}{6}$
Step-by-Step Solution
Key Concept: Apply L'Hôpital twice: $\lim_{x\to0}\frac{f(x)}{x^3}=\lim_{x\to0}\frac{f'(x)}{3x^2}=\lim_{x\to0}\frac{x+\sin(1-e^x)}{3x^2}$. Apply L'Hôpital again.
Three applications of L'Hôpital give $-\frac{1}{6}$.
Correct Answer: 1