Limits And Continuity — Formula Sheet

limₓ→ₐ (xⁿ − aⁿ)/(x − a) = n aⁿ⁻¹

limₓ→ₐ (xᵐ − aᵐ)/(xⁿ − aⁿ) = (m/n) aᵐ⁻ⁿ

limₓ→₀ (eˣ − 1)/x = 1

limₓ→₀ (aˣ − 1)/x = logₑ a

limₓ→₀ sin x / x = 1

limₓ→₀ (1 − cos x)/x² = 1/2

limₓ→ₐ sin(x − a)/(x − a) = 1

limₓ→₀ (1 + x)¹/ˣ = e

limₓ→∞ (1 + 1/x)ˣ = e

limₓ→₀ log(1 + x)/x = 1

limₓ→₀ (e^λˣ − 1)/x = λ

Standard limits like lim (x→0) sinx/x = 1 and lim (x→0) (eˣ - 1)/x = 1 are used frequently.