Download Lecture Slides on l'Hôpital's rule - Calculus II | MATH 211 and more Assignments Calculus in PDF only on Docsity! l’Hôpital’s Rule MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Summer 2008 J. Robert Buchanan l’Hôpital’s Rule Background Limits like the following two can be evaluated using clever algebraic and trigonometric arguments. 1 lim x→3 x2 − x − 6 x2 − 9 2 lim x→0 cos x − 1 x Today we explore a means of evaluating these and other limits more conveniently. J. Robert Buchanan l’Hôpital’s Rule Examples Example Use l’Hôpital’s Rule as appropriate to evaluate the following limits. 1 lim x→0 cos x − 1 x 2 lim x→π/2 1 − sin x 2 cos x 3 lim x→0 ex − e−x − 2 sin x x sin x 4 lim x→∞ x2 + 5x + 4 x ln x J. Robert Buchanan l’Hôpital’s Rule Generalized Mean Value Theorem In order to prove l’Hôpital’s Rule we will need the following. Theorem (Generalized Mean Value Theorem) Suppose f and g are continuous on [a, b] and differentiable on (a, b) and that g′(x) 6= 0 on interval (a, b). Then there exists a number z ∈ (a, b) such that f (b) − f (a) g(b) − g(a) = f ′(z) g′(z) . J. Robert Buchanan l’Hôpital’s Rule Proof of l’Hôpital’s Rule Proof. Suppose lim x→c f (x) g(x) is indeterminate of the form 0/0 and that lim x→c f ′(x) g′(x) = L. Define the following two functions. F (x) = { f (x) if x 6= c 0 if x = c and G(x) = { g(x) if x 6= c 0 if x = c J. Robert Buchanan l’Hôpital’s Rule Indeterminate form 00 Definition A limit of the form lim x→c f (x)g(x) where lim x→c f (x) = 0 and lim x→c g(x) = 0 is said to be indeterminate of the form 00. Note: If y = f (x)g(x) then ln y = g(x) ln(f (x)) and lim x→c g(x) ln(f (x)) is indeterminate of the form 0 · ∞. If lim x→c g(x) ln(f (x)) = L then lim x→c f (x)g(x) = eL. J. Robert Buchanan l’Hôpital’s Rule Example Example Evaluate the following limit. lim x→1− (1 − x)ln x J. Robert Buchanan l’Hôpital’s Rule Indeterminate form 1∞ Definition A limit of the form lim x→c f (x)g(x) where lim x→c f (x) = 1 and lim x→c g(x) = ±∞ is said to be indeterminate of the form 1∞. Note: If y = f (x)g(x) then ln y = g(x) ln(f (x)) and lim x→c g(x) ln(f (x)) is indeterminate of the form 0 · ∞. If lim x→c g(x) ln(f (x)) = L then lim x→c f (x)g(x) = eL. J. Robert Buchanan l’Hôpital’s Rule Example Example Evaluate the following limit. lim x→∞ (1 + ex)1/x J. Robert Buchanan l’Hôpital’s Rule Indeterminate form ∞−∞ Definition A limit of the form lim x→c (f (x) − g(x)) where lim x→c f (x) = ∞ and lim x→c g(x) = ∞ is said to be indeterminate of the form ∞−∞. Example Evaluate the following limit. lim x→∞ (sinh x − cosh x) J. Robert Buchanan l’Hôpital’s Rule Homework Read Section 3.2 and work exercises 1–51 odd. J. Robert Buchanan l’Hôpital’s Rule
