Limit of ((1+x)^(1/x) − e)/x

Short answer

Use series expansion: (1+x)^(1/x) = e^(ln(1+x)/x) = e·e^(−x/2+…) = e(1 − x/2 + …). Then the limit becomes −e/2.

Evaluate lim_{x→0} ((1+x)^(1/x) − e)/x.

Let L = lim_{x→0} [(1 + x)^(1/x) − e]/x. This is 0/0 since (1+x)^(1/x) → e.

Write (1 + x)^(1/x) = e^(ln(1+x)/x). Expand:

ln(1 + x)/x = 1 − x/2 + x²/3 − ...

(1 + x)^(1/x) = e^(1 − x/2 + x²/3 − ...) = e · e^(−x/2 + x²/3 − ...) = e[1 − x/2 + O(x²)]

So (1 + x)^(1/x) − e = −e x/2 + O(x²), and dividing by x:

Answer: L = −e/2.

(By L'Hospital: with y = (1+x)^(1/x), y′ = y[1/(x(1+x)) − ln(1+x)/x²]. The bracket tends to −1/2 as x → 0, giving y′ → −e/2.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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