$\lim_{x \to \infty} x \left[\left(1 + \frac{1}{x}\right)^x - e\right]$
Step-by-Step Solution
Key Concept: Write $(1 + 1/x)^x = e^{x \ln(1 + 1/x)}$. Expand $\ln(1 + 1/x) = 1/x - 1/(2x^2) + O(1/x^3)$, so exponent $= 1 - 1/(2x) + O(1/x^2)$. Then $(1 + 1/x)^x = e \cdot e^{-1/(2x) + \dots} \approx e(1 - 1/(2x))$. Thus $x[(1 + 1/x)^x - e] \to x \cdot (-e/(2x)) = -e/2$.
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Correct Answer: (3)