Find the limit of (x ^ (1 + x)) / ((1 + x) ^ x) - X / E at x positive infinity, | Math enthusiast | Mathematical art

Find the limit of (x ^ (1 + x)) / ((1 + x) ^ x) - X / E at x positive infinity,

X ^ (1 + x) / (1 + x) ^ x = x ^ X / (1 + x) ^ x * x = x / (1 + 1 / x) ^ x primitive = x [1 / (1 + 1 / x) ^ X - 1 / E] = x [e - (1 + 1 / x) ^ x] / [e (1 + 1 / x) ^ x] = x [e - (1 + 1 / x) ^ x] / e ^ 2 = 1 / e ^ 2 * [e - (1 + T) ^ (1 / T)] / T = 1 / e ^ 2 * [e - (E - (ET) / 2 + (11et ^ 2) / 24 - O (T ^ 3

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