Exponential Function

Definition: eˣ is the inverse of ln x Domain: D = ]−∞ ; +∞[ and eˣ > 0 always Graph of y = eˣ Algebraic Properties: e^(x+y), e^(x−y), e^(−x), e^(ln x) Equations & Inequalities: eᵃ = eᵇ ↔ a = b, eˣ = a ↔ x = ln a Derivatives: (eˣ)' = eˣ and (eᵘ)' = u'·eᵘ Key Limits: e^(−∞) = 0⁺, e^(+∞) = +∞ Growth Dominance: lim(x→+∞) eˣ/xⁿ = +∞ Decay: lim(x→−∞) xⁿeˣ = 0 Removing Indeterminate Forms: • ∞/∞ or 0/0 → L'Hôpital's Rule • 0 × ∞ → Change form + L'Hôpital (ex: xeˣ → x/e⁻ˣ) • ∞ − ∞ → Common factor