Limits
Limit involving $\sin x\cdot\sin2x\cdots\sin nx$ vs product form
MJMT_Full_Test_07
Grade 12
Question:
If $\displaystyle\lim_{x\to0}\frac{\sin x\sin2x\sin3x\cdots\sin nx-x^n}{x\tan((1+x)(1+2x)(1+3x)\cdots(1+2023x))}$ exists and is non-zero, then $n$ equals
2022
2023
2024
no such $n$
Step-by-Step Solution
Key Concept: Numerator: $\sin x\cdots\sin nx\sim x^n\cdot n!-x^n=x^n(n!-1)$ near 0 — non-zero if $n!\neq1$. Denominator: $x\tan(\prod(1+kx))\sim x\cdot(\prod(1+kx)-1)\sim x\cdot(\sum_{k=1}^{2023}kx)=x\cdot O(x)=O(x^2)$. For limit to exist (and be non-zero), degrees must match.
No such $n$ in options satisfies the condition (limit $=0$ for $n\geq2$, non-existent for $n<0$). Answer D.
Correct Answer: 4