Limit problem

(FB/Math/BM)

(This problem was inspired by a long (and continuing) discussion on products of primes but it stands alone.)

Find a function (or several or all) f(x) which satisfied all these three conditions:
(1) lim [x→∞] f(x)/aˣ = 0 if a ≥ e
…(2) f(x)/aˣ → ∞ when x→∞ if 0 < a < e
(3) lim [x→∞] f(x)1/x = e.

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Starting with f(x) = eˣ, conditions (2) and (3) are satisfied but not (1). Therefore, adjust a little so that f(x) grows slower by setting f(x) = exp(x – g(x)) where g(x) → ∞ but g(x)/x → 0 when x→∞ (so g(x) goes to infinity but ”slower” th…an x). Then

(1) lim [x→∞] f(x)/aˣ = lim [x→∞] eˣ/aˣ·exp(-g(x)) = 0,
since lim [x→∞] eˣ/aˣ ≤ 1 and lim [x→∞] exp(-g(x)) = 0.

(2) f(x)/aˣ = exp(x(1 – ln a) – g(x)) → ∞ when x→∞,
since x(1 – ln a) – g(x) → ∞, because ln a < 1.

(3) f(x)1/x = (eˣ)1/x·e-g(x)/x = e·e-g(x)/x → e when x→∞, since g(x)/x → 0.

Two simple examples are f(x) = exp(x – √x), the first I thought of, and f(x) = exp(x – ln x) = eˣ/x by Paul Stanford. – There is of course an infinity of possibilities like g(x) = xa for any a < 1, log x, log log x, … for any number of logs. But maybe the symptotic form must be exp(x – g(x)) where g(x) → ∞ and g(x)/x → 0 when x→∞. But that’s just a conjecture (or rather guess).

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