Question

Let \( a_n \) denote the term independent of \( x \) in the expansion of \[ \left[x + \frac{\sin(1/n)}{x^2}\right]^{3n}, \] then \[ \lim_{n\to\infty} \frac{(a_n)n!}{\,{}^{3n}P_n} \] equals:

• A. \( 0 \)
• B. \( 1 \)
• C. \( e \)
• D. \( \frac{e}{\sqrt{3}} \)

Hint:

For constant term in binomial: \begin{itemize} \item Match powers carefully. \item Use asymptotics like \( (1+1/n)^n \to e \). \end{itemize}

Answer 1

The Correct Option is C