分からない問題はここに書いてね435

log(1 + x) = x - (1/2)*x^2 + o(x^2) (x → 0)

log(1 + 1/x) = 1/x - (1/2)*(1/x^2) + o(1/x^2) (x → ∞)

x*log(1 + 1/x) = 1 - (1/2)*(1/x) + o(1/x) (x → ∞)

x*{(1 + 1/x)^x - e} = e*{exp(x*log(1 + 1/x) - 1) - 1} / (1/x)

exp(x*log(1 + 1/x) - 1) = 1 + (x*log(1 + 1/x) - 1) + o(x*log(1 + 1/x) - 1) (x → ∞)


[o(x*log(1 + 1/x) - 1) / (x*log(1 + 1/x) - 1)] * [(x*log(1 + 1/x) - 1) / (1/x)] → 0 * (-1/2) = 0 (x → ∞)

だから

o(x*log(1 + 1/x) - 1) = o(1/x)

exp(x*log(1 + 1/x) - 1) = 1 + (x*log(1 + 1/x) - 1) + o(1/x) (x → ∞)

[exp(x*log(1 + 1/x) - 1) - 1] / (1/x) = [(x*log(1 + 1/x) - 1) + o(1/x)] / (1/x) (x → ∞)

[exp(x*log(1 + 1/x) - 1) - 1] / (1/x) = [-(1/2)*(1/x) + o(1/x) + o(1/x)] / (1/x) (x → ∞)

[exp(x*log(1 + 1/x) - 1) - 1] / (1/x) → -1/2 (x → ∞)

x*{(1 + 1/x)^x - e} = e*{exp(x*log(1 + 1/x) - 1) - 1} / (1/x) → -(1/2)*e (x → ∞)