Evaluate Function's Limits Using l'Hospital RuleIntegration Techniques and Improper IntegralIntegration TechniquesEvaluate the Improper IntegralFind a Formula for General Term of a SequenceDetermine the Monotonicity of SequencesBoundedness of SequencesM408D: Class Outlines for l’Hospital Rule,Integrals, and SequencesReview and Problem Solving SkillsQimou [email protected], University of Te xas at AustinJune 24, 20041 Evaluate Function’s Limits Using l’Hospital RuleA Recipe for “Cooking” Limits using l’Hospital Rule.• Step 1: Recognize the 7 indeterminate forms:00,∞∞, 0 · ∞, ∞ − ∞, ∞0, 00, and 1∞• Step 2: Convert to the basic indeterminate types:00or∞∞• Step 3: Apply the l’Hospital Rule to evaluate the limit. (Taking derivatives on both numeratorand denominator)• Step 4: Check if the limit is determinate. If so, the food is ready. If the limit is still indeter-minate, goes to Step 1.For more details and examples, refer to my class notes of 06/15/04.2 Integration Techniques and Improper Integral2.1 Integration Techniques• Algebraic Skills: try to decompose or expand the integrand into small pieces and integratethem separately• Substitution: simple substitution / formal substitution / Trigonometric substitution• Integration by Parts: use the order I.L.A.T.E. (Inverse Trigonometric, Log, Algebraic, Trigono-metric, Exp) to make the choice for u.Rudv = uv −Rvdu• Partial Fraction Decomposition: decompose the rational function into partial fractions andinte grate them separatelyRefer to my class notes on 06/17/04 for details.12 M408D: Class Outlines for l’Hospital Rule, Integrals, and Sequences3 Evaluate the Improper IntegralProblem Type.1. Check if the integrand has irregular points (Is the integrand unbounded?)2. Find the anti-derivative (indefinite integral) using the integration techniques listed above3. Evaluate the definite integral (using Fundamental Theorem of Calculus)4. Take limit to evaluate the improper integralRefer to section 8.8 of textb ook and my c lass notes for details.4 Find a Formula for General Term of a SequenceProblem Type 1. Pattern recognition (like problem 9-14 on the textbook, P.747.) (Refer to myclass note on 06/22/04 for details)Problem Type 2. Find a formula for general term for sequence represented by a recursive equa-tion. e.g. suppose se quence ansatisfies that a1= 3 and an+1= 2an− 1. Try to find a formula forgeneral term an. (Refer to my class note on 06/22/04 for the solving skills)5 Determine the Monotonicity of SequencesHere are some effective skills for solving this kind of problems:• Using the monotonicity of functions. (good for simple an. For complex bn, check the sign ofthe derivative.) e.g. an= ln n or bn= ne−n• Compare the difference an+1− anto 0. (good for Algebraic an) e.g. an= n +1n• Compare the ratio an+1/anto 1, if an> 0 for all n. (good for factorial and exponential an)e.g. an=n!nnRefer to my class notes on 06/22/04 for details.6 Boundedness of SequencesRefer to my class notes on 06/22/04 for
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