L'Hopital's Rule for Indeterminate Forms: 0/0, ∞/∞, 0·∞, and ∞−∞ - Prof. M. Girardi, Study notes of Mathematics

Download L'Hopital's Rule for Indeterminate Forms: 0/0, ∞/∞, 0·∞, and ∞−∞ - Prof. M. Girardi and more Study notes Mathematics in PDF only on Docsity! Indeterminate Forms - L’Hôpital’s Rule At x = u, has the indeterminate form if lim x→u f(x) = and lim x→u g(x) = (1) f(x) g(x) 0 0 0 0 (2) f(x) g(x) ∞ ∞ ∞ ∞ (3) f(x) · g(x) 0 · ∞ 0 ∞ (4) f(x) − g(x) ∞−∞ ∞ ∞ (5) [f(x)]g(x) 00 0 0 (6) [f(x)]g(x) ∞0 ∞ 0 (7) [f(x)]g(x) 1∞ 1 ∞ HERE: u stands for any of the symbols a, a−, a+, −∞, +∞. L’Hôpital’s Rule (1) and (2) If: • f(x) g(x) has the interdeterminate form 0 0 or ∞∞ at u and • lim x→u f ′(x) g′(x) exists (i.e. this limit is a finite number or −∞ or ∞) then lim x→u f(x) g(x) = lim x→u f ′(x) g′(x) . (3) If f(x) · g(x) has the interdeterminate form 0 · ∞ at u, then rewrite: f(x) · g(x) = f(x) 1/g(x) , which has the interdeterminate form 0 0 at u or f(x) · g(x) = g(x) 1/f(x) , which has the interdeterminate form ∞ ∞ at u and then apply L’Hôpital’s Rule. 1