Limits and Improper Integrals: L'Hopital's Rule and Unbounded Intervals, Study notes of Calculus

Download Limits and Improper Integrals: L'Hopital's Rule and Unbounded Intervals and more Study notes Calculus in PDF only on Docsity! Math166 Section T Spring 2009 Review - Chapter 8 8.1 & 8.2 Indeterminate forms 1) 0 0 , ∞ ∞ forms: use the L’Hôpital’s rule.(several times, if needed) If f(c) g(c) = 0 0 or ∞ ∞ , then lim x→c f(x) g(x) = lim x→c f ′(x) g′(x) . 2) 0 · ∞, ∞−∞ forms: transform problems to 0 0 or ∞ ∞ , then use the L’Hôpital’s rule. 3) 00, ∞0, 1∞ forms: consider their logarithm, then find the limit of log of the given expression using the L’Hôpital’s rule, then find the limit of the given expression, that is, y = the given expression =⇒ lim x→c ln y =⇒ lim x→c y = lim x→c eln y Example: lim x→0+ xx. This is of the form 00. Now we proceed as follows: Let y = xx, then ln y = ln xx = x ln x. We first find the limit for ln y: lim x→0+ ln y = lim x→0+ x ln x which is in the form 0 ·∞. Write it as lim x→0+ ln x 1/x which is in the form ∞∞ . By L’Hopital’s Rule this limit is 0. Then, passing the limit: lim x→0+ y = lim x→0+ eln y = elimx→0+ ln y = e0 = 1. 4) Determinate forms 0 ∞ = 0, ∞ 0 = ±∞, 0 + 0 = 0, ∞+∞ =∞, 0 · 0 = 0, ∞ ·∞ =∞, 0∞ = 0, ∞∞ =∞ 8.3 Improper Integrals with Infinite Limits These are definite integrals with unbounded intervals. 1) ∫ b −∞ f(x) dx = lim a→−∞ ∫ b a f(x) dx and ∫ ∞ a f(x) dx = lim b→∞ ∫ b a f(x) dx If the limit exists and is finite, we say that the integral converges, otherwise it diverges. 1