2.5 Limits at Infinity

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1 Section 25 Limits at Infinity 1 25 Limits at Infinity Limits at infinity as opposed to infinite limits occur when the independent variable becomes large in magnitude For this reason, limits at infinity determine what is called the end behavior of a function An application of these limits is to determine whether a system (such as an ecosystem or a large oscillating structure) reaches a steady state as time increases Limits at Infinity and Horizontal Asymptotes Infinite Limits at Infinity End Behavior Quick Quiz SECTION 25 EXERCISES Review Questions 1 Explain the meaning of lim fhxl=10 2 What is a horizontal asymptote? fhxl 3 Determine lim if fhxlø100,000 and ghxlø as xø xø ghxl 4 Describe the end behavior of ghxl=e -2 x 5 Describe the end behavior of fhxl=-2 x 3 6 The text describes four cases that arise when examining the end behavior of a rational function fhxl= phxlêqhxl Describe the end behavior associated with each case 7 Evaluate lim xø e x, lim ex, and lim xø e -x 8 Use a sketch to find the end behavior of f HxL = ln x Basic Skills 9 14 Limits at infinity Evaluate the following limits 9 lim xø x 2 10 lim xø 11 lim qø cosq q x + 10 x 2 12 lim xø 3+2x+4x2 x 2 Copyright 2014 Pearson Education, Inc

2 2 Chapter 2 Limits cos 13 lim x5 xø x 14 lim sin4 x 3 x x Infinite limits at infinity Determine the following limits 15 lim xø x lim 3 x11 17 lim xø x lim x lim I3 x 12-9 x 7 M xø 20 lim I3 x7 + x 2 M 21 lim I-3 x16 + 2M 22 lim 2 x-8 23 lim xø I-12 x -5 M 24 lim I2 x x 3 M Rational functions Determine lim fhxl and lim fhxl for the following rational functions Then give the horizontal asymptote of f (if any) 4 x 25 fhxl= 20 x+1 26 fhxl= 3 x2-7 x x 27 fhxl= 6 x2-9 x+8 3 x x fhxl= 8 x x+2 29 fhxl= 3 x3-7 x x 2 Copyright 2014 Pearson Education, Inc

3 Section 25 Limits at Infinity 3 x fhxl= x 5 + x 2 - x 31 fhxl= 2 x+1 3 x fhxl= 12 x8-3 3 x 8-2 x 7 33 fhxl= 40 x5 + x 2 16 x 4-2 x 34 fhxl= -x x+8 T Slant (oblique) asymptotes Complete the following steps for the given functions a Use polynomial long division to find the slant asymptote of f b Find the vertical asymptotes of f a Graph f and all of its asymptotes with a graphing utility Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph 35 fhxl= x2-3 x+6 36 fhxl= x2-1 x+2 37 fhxl= x2-2 x+5 3 x fhxl= 3 x2-2 x+7 2 x-5 39 fhxl= 4 x3 + 4 x x+ 4 1+x 2 40 fhxl= 3 x2-2 x+5 3 x Algebraic functions Determine lim fhxl and lim fhxl for the following functions Then give the horizontal asymptote(s) of f (if any) 41 fhxl= 4 x x x fhxl= x x+ 1 Copyright 2014 Pearson Education, Inc

4 4 Chapter 2 Limits 43 fhxl= 3 x x x fhxl=4x 3 x- 9 x Transcendental functions Determine the end behavior of the following transcendental functions by evaluating appropriate limits Then provide a simple sketch of the associated graph, showing asymptotes if they exist 45 fhxl=-3 e -x 46 fhxl=2 x 47 fhxl=1-ln x 48 fhxl= ln x 49 fhxl=sin x 50 fhxl= 50 e 2 x Further Explorations 51 Explain why or why not Determine whether the following statements are true and give an explanation or counterexample a The graph of a function can never cross one of its horizontal asymptotes b A rational function f can have both lim xø fhxl= L (where L is finite) and lim fhxl= c The graph of any function can have at most two horizontal asymptotes Horizontal and vertical asymptotes a Analyze lim fhxl and lim fhxl, and then identify any horizontal asymptotes b Find the vertical asymptotes For each vertical asymptote x = a, evaluate lim fhxl and lim fhxl xøa- xøa + 52 fhxl= x2-4 x+3 x-1 53 fhxl= 2 x x x x x 2 54 fhxl= 16 x x 2 + x 2 2 x fhxl= 3 x4 + 3 x 3-36 x 2 x 4-25 x fhxl=16 x 2 4 x 2-16 x Copyright 2014 Pearson Education, Inc

5 Section 25 Limits at Infinity 5 57 fhxl= x2-9 xhx-3l 58 fhxl= x-1 x 2ê fhxl= x x+ 6-3 x-1 1-x 2 60 fhxl= xhx+1l 61 fhxl= x - x End behavior for transcendental functions 62 The central branch of fhxl=tan x is shown in the figure a Evaluate lim tan x and lim tan x Are these limits infinite limits or limits at infinity? xøpê2- xøpê2 + b Sketch a graph of ghxl=tan -1 x by reflecting the graph of f over the line y= x, and use it to evaluate lim xø tan-1 x and lim tan-1 x 63 Consider the graph of y= sec -1 x (see Section 14) and evaluate the following limits using the graph Assume the domain is 8x : x 1< a lim xø sec -1 x b lim sec-1 x 64 The hyperbolic cosine function, denoted cosh x, is used to model the shape of a hanging cable (a telephone wire, for example) It is defined as cosh x= ex + e -x 2 a Determine its end behavior by evaluating lim cosh x and lim cosh x b Evaluate cosh 0 Use symmetry and part (a) to sketch a plausible graph for y = cosh x Copyright 2014 Pearson Education, Inc

6 6 Chapter 2 Limits 65 The hyperbolic sine function is defined as sinh x= ex - e -x 2 a Determine its end behavior by evaluating lim sinh x and lim sinh x b Evaluate sinh 0 Use symmetry and part (a) to sketch a plausible graph for y = sinh x Sketching graphs Sketch a possible graph of a function f that satisfies all of the given conditions Be sure to identify all vertical and horizontal asymptotes 66 fh-1l=-2, fh1l=2, fh0l= 0, lim xø fhxl=1, 67 lim xø0 + fhxl=, lim fhxl=-, lim fhxl=1, xø0- xø lim fhxl=-1 lim fhxl=-2 68 Asymptotes Find the vertical and horizontal asymptotes of fhxl=e 1êx 69 Asymptotes Find the vertical and horizontal asymptotes of fhxl= cos x + 2 x x Applications Steady states If a function f represents a system that varies in time, the existence of lim fhtl means that the tø system reaches a steady state (or equilibrium) For the following systems, determine if a steady state exists and give the steady-state value 70 The population of a bacteria culture is given by phtl= 2500 t The population of a culture of tumor cells is given by phtl= 3500 t t+1 72 The amount of drug (in milligrams) in the blood after an IV tube is inserted is mhtl=200i1-2 -t M 73 The value of an investment is given by vhtl=$1000 e 0065 t 74 The population of a colony of squirrels is given by phtl = e -01 t 75 The amplitude of an oscillator is given by ahtl=2 t + sin t t Looking ahead to sequences A sequence is an infinite ordered list of numbers that is often defined by a function For example, the sequence 82, 4, 6, 8, < is specified by the function fhnl= 2 n, where n=1, 2, 3, The limit of such a sequence is lim nø fhnl, provided the limit exists All the limit laws for limits at infinity may be applied to limits of sequences Find the limit of the following sequences, or state that the limit does not exist 76 :4, 2, 4, 1, 4, 2, >, which is defined by fhnl= 4 for n= 1, 2, 3, n 77 :0, 1 2, 2 3, 3 n- 1, >, which is defined by fhnl= for n=1, 2, 3, 4 n Copyright 2014 Pearson Education, Inc

7 Section 25 Limits at Infinity 7 78 : 1 2, 4 3, 9 4, 16 n2, >, which is defined by fhnl= for n=1, 2, 3, 5 n+1 79 :2, 3, 4, 4 9 Additional Exercises 5, >, which is defined by fhnl= n+ 1 for n=1, 2, 3, 16 n 2 80 End behavior of rational functions Suppose fhxl= phxl is a rational function, where qhxl phxl=a m x m + a m-1 x m-1 + +a 2 x 2 + a 1 x+a 0 and qhxl=b n x n + b n-1 x n-1 + +b 2 x 2 + b 1 x+b 0, a m 0, and b n 0 a Prove that if m = n, then lim xø fhxl= a m b n b Prove that if m < n, then lim fhxl= 0 xø 81 Horizontal and slant asymptotes a Is it possible for the graph of a rational function to have both slant and horizontal asymptotes? Explain b Is it possible for analgebraic function to have two different slant asymptotes? Explain or give an example T T 82 End behavior of exponentials Use the following instructions to evaluate limits of fhxl= ex + e 2 x a Evaluate lim xø fhxl by dividing both the numerator and denominator by e 3 x b Evaluate lim fhxl by dividing both the numerator and denominator by e 2 x c Give the horizontal asymptote(s) d Graph f to confirm your work in parts (a)-(c) e 2 x + e 3 x Limits of exponentials Evaluate lim fhxl and lim fhxl Then give the horizontal asymptote(s) of f Confirm your findings by plotting f 83 fhxl= 2 ex + 3 e 2 x e 2 x + e 3 x 84 fhxl= 3 ex + e -x e x + e -x lni9- x 2 M T 85 Subtle asymptotes Use analytical methods to identify all the asymptotes of f HxL = Then confirm your 2 e x - e-x results by locating the asymptotes using a graphing calculator Copyright 2014 Pearson Education, Inc