Suppose that a := limx→∞(1 + 1/x)x exists and is greater than 1 (see Example). Assume that...

Suppose that a := limx→∞(1 + 1/x)x exists and is greater than 1 (see Example). Assume that ax : R → (0,∞) is onto, continuous, strictly increasing, and satisfies axay = ax+y and (ax)y = axy for all x, y ∈ R (see Exercise). Let L(x) denote the inverse function of ax .

a) Prove that t L(1 + 1/t) → 1 as t →∞.

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b) Prove that (ah − 1)/h → 1 as h → 0.

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c) Prove that ax is differentiable on R and (ax)′ = ax for all x ∈ R.

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d) Prove that L′(x) = 1/x for all x > 0.

[Note: a is the natural base e and L(x) is the natural logarithm log x.]

EXAMPLE

Prove that the sequence (1 + 1/n)n is increasing, as n → ∞, and its limit L satisfies 2 < L ≤ 3. (The limit L turns out to be an irrational number, the natural base e = 2.718281828459 · · · .)