Rankwise

limx→∞ (1 + a/x)ˣ = eᵃ

The limit of (1 + a/x)ˣ as x tends to ∞ is eᵃ.

Proof

  1. Write (1 + a/x)ˣ = [(1 + a/x)^(x/a)]ᵃ.
  2. The bracket tends to e as x/a → ∞, so the limit is eᵃ.
  3. Shortcut for any 1^∞ form: lim f^g = e^(lim g(f − 1)). Here g(f − 1) = x × a/x = a.

Standard limits

LimitValue
limx→0 sin x / x1
limx→0 tan x / x1
limx→0 (1 − cos x) / x²1/2
limx→0 (1 − cos x) / x0
limx→0 (eˣ − 1) / x1
limx→0 (aˣ − 1) / xlog a
limx→0 log(1 + x) / x1
limx→0 (1 + x)^(1/x)e
limx→∞ (1 + 1/x)ˣe
limx→∞ (1 + a/x)ˣeᵃ
limx→a (xⁿ − aⁿ) / (x − a)n aⁿ⁻¹
limx→0 sin⁻¹x / x1
limx→0 tan⁻¹x / x1
limx→∞ sin x / x0
limx→0⁺ x log x0
limx→0⁺ xˣ1

Questions students ask

What is the limit of (1 + a/x)ˣ as x tends to ∞?

eᵃ.

How do you prove lim x→∞ (1 + a/x)ˣ = eᵃ?

Write (1 + a/x)ˣ = [(1 + a/x)^(x/a)]ᵃ. The bracket tends to e as x/a → ∞, so the limit is eᵃ. Shortcut for any 1^∞ form: lim f^g = e^(lim g(f − 1)). Here g(f − 1) = x × a/x = a.

Use these next

Standard limitssin x/x, (eˣ − 1)/x, (1 + x)^(1/x) and every other standard limit, with proofs.Derivative formulasEvery standard derivative with its proof.Integration formulasEvery standard integral with a step-by-step proof.Maths formula sheetAlgebra to integrals, the formulas JEE Main uses most.