Math 1351 011 November 2 2007 1 Announcements Homework 9 due this Monday 11 5 2007 Exam 3 will be Monday 11 19 2007 TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 2 Limits to Infinity lim f x L x means that f x approaches the number L as x increases without bound lim f x M x means that f x approaches the number M as x decreases without bound TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 3 Infinite Limits lim f x x c means that f increases without bound as x approaches c from either side lim g x x c means that g decreases without bound as x approaches c from either side TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 4 Asymptotes The line x c is a vertical asymptote of the graph of f if either of the one sided limits lim f x or x c lim f x x c is infinite or The line y L is a horizontal asymptote of the graph of f if lim f x L x or lim f x L x TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 5 Vertical tangents and cusps Suppose the function f is continuous at the point P c f c Then the graph of f has a vertical tangent at P if limx c f 0 x and limx c f 0 x are either both or both a cusp at P if limx c f 0 x and limx c f 0 x are both infinite with opposite signs one and the other TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 6 Summary of Graphing Strategy Simplify the equation as much as possible Find 1st derivatives and critical numbers Determine intervals of increase and decrease Apply 2nd derivative test Determine concavity and points of inflection Apply 1st derivative test Find asymptotes vertical tangents cusps Plot points Sketch the curve See detailed summary on last page of section 4 3 of Strauss Page 226 of 5th edition TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 7 L Ho pital s Rule Let f and g be differentiable functions with g 0 x 6 0 on an open interval containing c except possibly c itself Suppose f x lim x c g x produces an indeterminate form 0 0 or and that f 0 x lim L x c g 0 x where L is either a finite number or Then f x L lim x c g x TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 8 L Ho pital s Rule also applies to one sided limites and to limits to infinity i e where x goes to or Be careful to check that the limit gives an indeterminate form before applying L Ho pital s Rule Applying it blindly can give you the wrong answer TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 9 Note that L Ho pital s Rule applies only to the indeterminate forms 00 and For other indeterminate forms such as 1 00 0 and 0 need to manipulate into the form 00 or then evaluate with L Ho pital s Rule TTU Department of Mathematics Statistics Math 1351 011 November 2 2007 10 Special limits with ex and ln x If k and n are positive numbers then ln x lim x 0 xn ekx lim x xn ln x lim 0 x xn lim xn e kn 0 x TTU Department of Mathematics Statistics
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