Limits, Continuity & Differentiability
Continuity at Zero — Series Expansion
nta_pyq_2024_apr
Grade 12
Question:
If the function $f(x)=\dfrac{\sin3x+\alpha\sin x-\beta\cos3x}{x^3}$, $x\in\mathbb{R}$, is continuous at $x=0$, then $f(0)$ is equal to:
Step-by-Step Solution
Key Concept: Expand using Taylor series. For the limit to exist (finite), coefficients of $x^{-3}$ and $x^{-1}$ must vanish. Constant term gives $f(0)$.
$\beta=0$, $\alpha=-3$. $f(0)=\frac{-27+3}{6}=-4$.
Correct Answer: 4