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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
A step function Let f(x)=xx, for x 0. a. Sketch a graph of f on the interval [ 2, 2]. b. Does limx0f(x) exist ? Explain your reasoning after first examining limx0f(x) and limx0+f(x).The floor function For any real number x, the floor function (or greatest integer function) x is the greatest integer less than or equal to x (see figure). a. Compute limx1x, limx1+x, limx2x, and limx2+x. b. Compute limx2.3x, limx2.3+x, and limx2.3x. c. For a given integer a, state the values of limxax and limxa+x. d. In general, if a is not an integer, state the values of limxax and limxa+x. e. For what values of a does limxax exist? Explain.The ceiling function For any real number x, the ceiling function x is the smallest integer greater than or equal to x. a. Graph the ceiling function y = x, for 2 x 3. b. Evaluate limx2x, limx1+x, and limx1.5x. c. For what values of a does limxax exist? Explain.Limits of even functions A function f is even if f(x) = f(x), for all x in the domain of f. Suppose f is even, with limx2+f(x)=5 and limx2f(x)=8. Evaluate the following limits. a. limx2+f(x) b. limx2f(x)Limits of odd functions A function g is odd if g(x) = g(x), for all x in the domain of g. Suppose g is odd, with limx2+g(x)=5 and limx2g(x)=8. Evaluate the following limits. a. limx2+g(x) b. limx2g(x)Limits by graphs a. Use a graphing utility to estimate limx0tan2xsinx, limx0tan3xsinx, and limx0tan4xsinx. b. Make a conjecture about the value of limx0tanpxsinx, for any real constant p.Limits by graphs Graph f(x)=sinnxx, for n = 1, 2, 3, and 4 (four graphs). Use the window [1, 1] [0, 5]. a. Estimate limx0sinxx, limx0sin2xx, limx0sin3xx, and limx0sin4xx. b. Make a conjecture about the value of limx0sinpxx, for any real constant p.Limits by graphs Use a graphing utility to plot y=sinpxsinqx for at least three different pairs of nonzero constants p and q of your choice. Estimate limx0sinpxsinqx in each case. Then use your work to make a conjecture about the value of limx0sinpxsinqx for any nonzero values of p and q.Use Theorem 2.4 to evaluate limx2(2x48x16) and limx1x1x.Use Theorem 2.4 to compute limx15x43x2+8x6x+1.Evaluate limx5x27x+10x5.Suppose f satisfies 1f(x)1+x26 for all values of x near zero. Find limx0f(x), if possible.How is limxap(x) calculated if p is a polynomial function?Evaluate limx1(x3+3x23x+1).For what values of a does limxar(x)=r(a) if r is a rational function?Evaluate limx4(x44x13x1).Explain why limx3x27x+12x3=limx3(x4) and then evaluate limx3x27x+12x3.Evaluate limx5(4x2100x5).Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits and stale the limit laws used to justify your computations. 17. limx1(4f(x))Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits and stale the limit laws used to justify your computations. 18. limx1f(x)h(x)Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits and stale the limit laws used to justify your computations. 19. limx1(f(x)g(x))Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits and stale the limit laws used to justify your computations. 20. limx1(f(x))h(x))Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits and stale the limit laws used to justify your computations. 22. limx1f(x)g(x)h(x)Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits and stale the limit laws used to justify your computations. 24. limx1f(x)g(x)+33Assume limx1f(x)=8 limx1g(x)=3, and limx1h(x)=2. Compute the following limits and state the limit laws used to justify your computations. 13.limx1(f(x))2/3.How are limxap(x) and limxa+p(x) calculated if p is a polynomial function?Suppose g(x)={2x+1ifx05ifx=0. Compute g(0) and limx0g(x).Suppose f(x)={4ifx3x+2ifx3. Compute limx3f(x) and limx3+f(x).Suppose p and q are polynomials. If limx0p(x)q(x)=10 and q(0) = 2, find p(0).Suppose limx2f(x)=limx2h(x)=5. Find limx2g(x), where f(x) g(x) h(x), for all x.Limits of linear functions Evaluate the following limits. 11. limx4(3x7)Limits of linear functions Evaluate the following limits. 12. limx1(2x+5)Limits of linear functions Evaluate the following limits. 13. limx95xLimits of linear functions Evaluate the following limits. 15. limx64Evaluating limits Evaluate the following limits. 25. limx1(2x33x2+4x+5)Evaluating limits Evaluate the following limits. 26. limt2(t2+5t+7)Evaluating limits Evaluate the following limits. 27. limx15x2+6x+18x4Evaluating limits Evaluate the following limits. 28. limt3t2103Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 27.limp23p4p+11Evaluating limits Evaluate the following limits. 30. limx2(x2x)5Evaluating limits Evaluate the following limits. 31. limx35x4x3Evaluating limits Evaluate the following limits. limh0316+3h+4Evaluating limits Evaluate the following limits, where c and k are constants. 61. limx2(5x6)3/2Evaluating limits Evaluate the following limits, where c and k are constants. 60. limh0100(10h1)11+2Other techniques Evaluate the following limits, where a and b are fixed real numbers. 39. limx1x21x1Other techniques Evaluate the following limits, where a and b are fixed real numbers. 40. limx3x22x3x3Other techniques Evaluate the following limits, where a and b are fixed real numbers. 41. limx4x2164xOther techniques Evaluate the following limits, where a and b are fixed real numbers. 42. limt23t27t+22tOther techniques Evaluate the following limits, where a and b are fixed real numbers. 43. limxb(xb)50x+bxbOther techniques Evaluate the following limits, where a and b are fixed real numbers. 44. limxb(x+b)7+(x+b)104(x+b)Other techniques Evaluate the following limits, where a and b are fixed real numbers. 45. limx1(2x1)29x+1Other techniques Evaluate the following limits, where a and b are fixed real numbers. 46. limh015+h15hOther techniques Evaluate the following limits, where a and b are fixed real numbers. 47. limx9x3x9Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 42.limw1(1w2w1w1)Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 43.limt5(1t24t516(t5))Other techniques Evaluate the following limits, where a and b are fixed real numbers. 48. limt3((4t2t3)(6+tt2))Other techniques Evaluate the following limits, where a and b are fixed real numbers. 49. limxaxaxa,a0Other techniques Evaluate the following limits, where a and b are fixed real numbers. 50. limxax2a2xa,a0Other techniques Evaluate the following limits, where a and b are fixed real numbers. 51. limh016+h4hEvaluating limits Evaluate the following limits, where c and k are constants. 66. limxcx22cx+c2xcEvaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 49.limx41x14x4Evaluating limits Evaluate the following limits, where c and k are constants. 62. limx31x2+2x115x3Evaluating limits Evaluate the following limits, where c and k are constants. 63. limx110x91x1Evaluating limits Evaluate the following limits, where c and k are constants. 64. limx2(1x22x22x)Evaluating limits Evaluate the following limits, where c and k are constants. 65. limh0(5+h)225hEvaluating limits Evaluate the following limits, where c and k are constants. 67. limwkw2+5kw+4k2w2+kw, for k 0Limits involving conjugates Evaluate the following limits. 77. limx1x1x1Limits involving conjugates Evaluate the following limits. 78. limx1x14x+53Limits involving conjugates Evaluate the following limits. 79. limx43(x4)x+53x+5Limits involving conjugates Evaluate the following limits. 80. limx0xcx+11, where c is a nonzero constantEvaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 59.limx0xcosxEvaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 60.limx0sin2xsinxEvaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 61.limx01cosxcos2x3cosx+2Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 62.limx0cosx1cos2x1Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 63.limx0x2x|x|Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 64.limw3|w3|w27w+12Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 65.limt2+|2t4|t24Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 66.limx1g(x), where g(x)={x21x+1ifx12ifx1Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 67.limx3x3|x3|Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 68.limx5|x5|x225Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 69.limx1x3+1x1Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 70.limx1+x1x21Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers. a. If limxaf(x)=L, then f(a) = L. b. If limxaf(x)=L, then limxa+f(x)=L. c. If limxaf(x)=L and limxag(x)=L, then f(a) = g(a). d. The limit limxaf(x)g(x) does not exist if g(a) = 0. e. If limx1+f(x)=limx1+f(x), it follows that limx1f(x)=limx1f(x).One-sided limits Let g(x)={5x15ifx46x+1ifx4. Compute the following limits or state that they do not exist. a. limx4g(x) b. limx4+g(x) c. limx4g(x)One-sided limits Let f(x)={x2ifx1x+1ifx1. Compute the following limits or state that they do not exist. a. limx1f(x) b. limx1+f(x) c. limx1f(x)One-sided limits Let f(x)={0ifx525x2if5x53xifx5. Compute the following limits or state that they do not exist. a. limx5f(x) b. limx5+f(x) c. limx5f(x) d. limx5f(x) e. limx5+f(x) f. limx5f(x)One-sided limits a. Evaluate limx2+x2. b. Explain why limx2x2 does not exist.One-sided limits a. Evaluate limx3x32x. b. Explain why limx3+x32x does not exist.77ETorricellis Law A cylindrical tank is filled with water to a depth of 9 meters. At t = 0, a drain in the bottom of the tank is opened and water flows out of the tank. The depth of water in the tank (measured from the bottom of the tank) t seconds after the drain is opened is approximated by d(t) = (3 0.015t)2, for 0 200. Evaluate and interpret limt200d(t).Limit of the radius of a cylinder A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r(S)=12(100+2S10). Find limS0+r(S) and interpret your result.A problem from relativity theory Suppose a spaceship of length L0 travels at a high speed v relative to an observer. To the observer, the ship appears to have a smaller length given by the Lorentz contraction formula L=L01v2c2, where c is the speed of light. a. What is the observed length L of the ship if it is traveling at 50% of the speed of light? b. What is the observed length L of the ship if it is traveling at 75% of the speed of light? c. In parts (a) and (b), what happens to L as the speed of the ship increases? d. Find limvcL01v2c2 and explain the significance of this limit.Applying the Squeeze Theorem a. Show that xxsin1xx, for x 0. b. Illustrate the inequalities in part (a) with a graph. c. Use the Squeeze Theorem to show that limx0xsin1x=0.A cosine limit by the Squeeze Theorem It can be shown that 1x22cosx1, for x near 0. a. Illustrate these inequalities with a graph. b. Use these inequalities to evaluate limx0cosx.A sine limit by the Squeeze Theorem It can be shown that 1x26sinx1, for x near 0. a. Illustrate these inequalities with a graph. b. Use these inequalities to evaluate limx0sinxx.A logarithm limit by the Squeeze Theorem a. Draw a graph to verify that |x| x2 ln x2 |x|, for l x 1, where x 0. b. Use the Squeeze Theorem to evaluate limx0x2lnx2.Absolute value limit Show that limx0x=0 by first evaluating limx0x and limx0+x. Recall that x={xifx0xifx0.Absolute value limit Show that limxax=a, for any real number. (Hint: Consider the cases a 0 and a 0.)Finding a constant Suppose f(x)={x25x+6x3ifx3aifx=3. Determine a value of the constant a for which limx3f(x)=f(3).Finding a constant Suppose f(x)={3x+bifx2x2ifx2. Determine a value of the constant b for which limx2f(x) exists and state the value of the limit, if possible.Finding a constant Suppose g(x)={x25xifx1ax37ifx1. Determine a value of the constant a for which limx1g(x) exists and state the value of the limit, if possible.Useful factorization formula Calculate the following limits using the factorization formula xnan=(xa)(xn1+xn2a+xn3a2++xan2+an1), where n is a positive integer and a is a real number. 70. limx2x532x2Useful factorization formula Calculate the following limits using the factorization formula xnan=(xa)(xn1+xn2a+xn3a2++xan2+an1), where n is a positive integer and a is a real number. 71. limx1x61x1Useful factorization formula Calculate the following limits using the factorization formula xnan=(xa)(xn1+xn2a+xn3a2++xan2+an1), where n is a positive integer and a is a real number. 72. limx1x7+1x+1(Hint: Use the formula for x7 a7 with a = 1.)Useful factorization formula Calculate the following limits using the factorization formula xnan=(xa)(xn1+xn2a+xn3a2++xan2+an1), where n is a positive integer and a is a real number. 73. limxax5a5xaUseful factorization formula Calculate the following limits using the factorization formula xnan=(xa)(xn1+xn2a+xn3a2++xan2+an1), where n is a positive integer and a is a real number. 74. limxaxnanxa, for any positive integer nSlope of a tangent line a. Sketch a graph of y = 2x and carefully draw three secant lines connecting the points P(0, 1) and Q(x, 2x), for x = 3, 2, and 1. b. Find the slope of the line that passes through P(0, 1) and Q(x, 2x), for x 0. c. Complete the table and make a conjecture about the value of limx02x1x.96EEven function limits Suppose f is an even function where limx1f(x)=5 and limx1+f(x)=6. Find limx1f(x) and limx1+f(x).Odd function limits Suppose g is an even function where limx1g(x)=5 and limx1+g(x)=6. Find limx1g(x) and limx1+g(x).Useful factorization formula Calculate the following limits using the factorization formula xnan=(xa)(xn1+xn2a+xn3a2++xan2+an1), where n is a positive integer and a is a real number. 75. limx1x31x1(Hint: x1=(x3)3(1)3.)Evaluate limx16x42x16.Creating functions satisfying given limit conditions Find functions f and g such that limx1f(x)=0 and limx1(f(x)g(x))=5.Creating functions satisfying given limit conditions Find a function f satisfying limx1(f(x)x1)=2.Finding constants Find constants b and c in the polynomial p(x) = x2 + bx + c such that limx2p(x)x2=6. Are the constants unique?Limits of composite functions 88. If limx1f(x)=4, find limx1f(x2).105ETwo trigonometric inequalities Consider the angle in standard position in a unit circle, where 0 /2 or /2 0 (use both figures). a. Show that |AC| = |sin |, for /2 /2. (Hint: Consider the cases 0 /2 and /2 0 separately.) b. Show that |sin | ||, for /2 /2. (Hint: The length of arc AB is , if 0 /2, and , if /2 0.) c. Conclude that || sin ||, for /2 /2. d. Show that 0 1 cos ||, for /2 0 /2.107ESketch the graph of a function and its vertical asymptote that satisfies the conditions limx2+f(x)= and limx2f(x)=.Analyze limx0+x5x and limx0x5x by determining the sign of the numerator and denominatorVerify that x(x+4)0 through negative values as x4+.The line x = 2 is not a vertical asymptote of y=(x1)(x2)x2. Why not?Explain the meaning of limxa+f(x)Explain the meaning of limxaf(x)=.What is a vertical asymptote?Consider the function F(x) = f(x)/g(x) with g(a) = 0. Does F necessarily have a vertical asymptote at x = a? Explain your reasoning.Analyzing infinite limits numerically Compute the values of f(x)=x+1(x1)2 in the following table and use them to determine limx1f(x).Analyzing infinite limits graphically Use the graph of f(x)=x(x22x3)2 to determine limx1f(x) and limx3f(x).Analyzing infinite limits graphically The graph of f in the figure has vertical asymptotes at x = 1 and x = 2. Analyze the following limits. a. limx1f(x) b. limx1+f(x) c. limx1f(x) d. limx2f(x) e. limx2+f(x) f. limx2f(x)Analyzing infinite limits graphically The graph of g in the figure has vertical asymptotes at x = 2 and x = 4. Analyze the following limits. a. limx2g(x) b. limx2+g(x) c. limx2g(x) d. limx4g(x) e. limx4+g(x) f. limx4g(x)Analyzing infinite limits graphically The graph of h in the figure has vertical asymptotes at x = 2 and x = 3 Analyze the following limits. a. limx2h(x) b. limx2+h(x) c. limx2h(x) d. limx3h(x) e. limx3+h(x) f. limx3h(x)Analyzing infinite limits graphically The graph of p in the figure has vertical asymptotes at x = 2 and x = 3. Analyze the following limits. a. limx2p(x) b. limx2+p(x) c. limx2p(x) d. limx3p(x) e. limx3+p(x) f. limx3p(x)Analyzing infinite limits graphically Graph the function f(x)=1x2x using a graphing utility with the window [1, 2] [10, 10]. Use your graph to determine the following limits. a. limx0f(x) b. limx0+f(x) c. limx1f(x) d. limx1+f(x)Analyzing infinite limits graphically Graph the function f(x)=exx(x+2)2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits. a. limx2+f(x) b. limx2f(x) c. limx0f(x) d. limx0+f(x)Suppose f(x) 100 and g(x) 0, with g(x) 0, as x 2. Determine limx2f(x)g(x).Evaluate limx31x3 and limx3+1x3.Verity that the function f(x)=x24x+3x23x+2 is undefined at x = 1 and at x = 2. Does the graph of f have vertical asymptotes at both these values of x? ExplainEvaluate limx0x+11cosx.Sketching graphs Sketch a possible graph of a function f, together with vertical asymptotes, satisfying all the following conditions on [0, 4]. f(1)=0,f(3)isundefined,limx3f(x)=1,limx0+f(x)=,limx2f(x)=,limx4f(x)=Sketching graphs Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions. g(2)=1,g(5)=1,limx4g(x)=,limx7g(x)=,limx7+g(x)=Which of the following statements are correct? Choose all that apply. a. limx11(x1)2 does not exist b. limx11(x1)2= c. limx11(x1)2=Which of the following statements are correct? Choose all that apply. a. limx1+11x does not exist b. limx1+11x= c. limx1+11x=Determining limits analytically Determine the following limits or state that they do not exist. 17. a. limx2+1x2 b. limx21x2 c. limx21x2Determining limits analytically Determine the following limits or state that they do not exist. 18. a. limx3+2(x3)3 b. limx32(x3)3 c. limx32(x3)3Determining limits analytically Determine the following limits or state that they do not exist. 19. a. limx4+x5(x4)2 b. limx4x5(x4)2 c. limx4x5(x4)2Determining limits analytically Determine the following limits. 24.a.limx1+x|x1| b.limx1x|x1| c.limx1x|x1|Determining limits analytically Determine the following limits. 25.a.limz3+(z1)(z2)(z3) b.limz3(z1)(z2)(z3) c.limz3(z1)(z2)(z3)Determining limits analytically Determine the following limits or state that they do not exist. 22. a. limx2+(x4)x(x+2) b. limx2(x4)x(x+2) c. limx2(x4)x(x+2)Determining limits analytically Determine the following limits or state that they do not exist. 23. a. limx2+x24x+3(x2)2 b. limx2x24x+3(x2)2 c. limx2x24x+3(x2)2Determining limits analytically Determine the following limits. 28.a.limt2+t35t2+6tt44t2 b.limt2t35t2+6tt44t2 c.limt2t35t2+6tt44t2 d.limt2t35t2+6tt44t2Determine limits analytically Determine the following limits. 29.a. limx2+1x(x2) b. limx21x(x2) c. limx21x(x2)Determine limits analytically Determine the following limits. 30.a. limx1+x3x25x+4 b. limx1x3x25x+4 c. limx1x3x25x+4Determine limits analytically Determine the following limits. 31.a. limx0x3x49x2 b. limx3x3x49x2 c. limx3x3x49x2Determine limits analytically Determine the following limits. 32.a. limx0x2x54x3 b. limx2x2x54x3 c. limx2x2x54x3Determining limits analytically Determine the following limits or state that they do not exist. 25. limx0x35x2x2Determining limits analytically Determine the following limits or state that they do not exist. 26. limt54t2100t5Determining limits analytically Determine the following limits or state that they do not exist. 27. limx1+x25x+6x1Determining limits analytically Determine the following limits or state that they do not exist. 28. limz4z5(z210z+24)2Determine limits analytically Determine the following limits. 37.limx6+x7x6Determine limits analytically Determine the following limits. 38.limx2x1(x3)(x2)Trigonometric limits Determine the following limits. 35. lim0+cscTrigonometric limits Determine the following limits. 36. limx0cscxTrigonometric limits Determine the following limits. 37. limx0+(10cotx)Trigonometric limits Determine the following limits. 38. lim/2+13tanDetermine limits analytically Determine the following limits. 43.lim02+sin1cos2Determine limits analytically Determine the following limits. 44.lim0sincos21Location of vertical asymptotes Analyze the following limits and find the vertical asymptotes of f(x)=x5x225. a. limx5f(x) b. limx5f(x) c. limx5+f(x)Location of vertical asymptotes Analyze the following limits and find the vertical asymptotes of f(x)=x+7x449x2. a. limx7f(x) b. limx7+f(x) c. limx7f(x) d. limx0f(x)Finding vertical asymptotes Find all vertical asymptotes x = a of the following functions. For each value of a, determine limxa+f(x),limxaf(x), and limxaf(x). 31. f(x)=x29x+14x25x+6Finding vertical asymptotes Find all vertical asymptotes x = a of the following functions. For each value of a, determine limxa+f(x),limxaf(x), and limxaf(x). 32. f(x)=cosxx2+2xFinding vertical asymptotes Find all vertical asymptotes x = a of the following functions. For each value of a, determine limxa+f(x),limxaf(x), and limxaf(x). 33. f(x)x+1x34x2+4xFinding vertical asymptotes Find all vertical asymptotes x = a of the following functions. For each value of a, determine limxa+f(x),limxaf(x), and limxaf(x). 34. f(x)x310x2+16xx28xAnalyzing infinite limits graphically Graph the function y = tan x with the window [, ] [10, 10]. Use the graph to analyze the following limits. a. limx/2+tanx b. limx/2tanx c. limx/2+tanx d. limx/2tanxAnalyzing infinite limits graphically Graph the function y = sec x tan x with the window [, ] [10, 10]. Use the graph to analyze the following limits. a. limx/2+secxtanx b. limx/2secxtanx c. limx/2+secxtanx d. limx/2secxtanxExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The line x = 1 is a vertical asymptote of the function f(x)=x27x+6x21. b. The line x = 1 is a vertical asymptote of the function f(x)=x27x+6x21. c. If g has a vertical asymptote at x = 1 and limx1+g(x)=, then limx1g(x)=.Matching Match functions af with graphs AF in the figure without using a graphing utility. a. f(x)=xx2+1 b. f(x)=xx21 c. f(x)=1x21 d. f(x)=x(x1)2 e. f(x)=1(x1)2 f. f(x)=xx+1Finding a rational function Find a rational function r(x) such that r(1) is undefined, limx1r(x)=0, and limx2r(x)=.Finding a function with vertical asymptotes Kind polynomials p and q such that f = p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.Finding a function with infinite limits Give a formula for a function f that satisfies limx6+f(x)= and limx6f(x)=.Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 58.f(x)=x21x41Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 45. f(x)x23x+2x10x9Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 46. g(x) = 2 ln x2Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 47. h(x)=ex(x+1)3Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 48. p(x)=sec(x2),forx2Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 49. g()=tan10Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 50. q(s)=ssins65EAsymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 52. g(x) = e1/xLimits with a parameter Let f(x)=x27x+12xa. a. For what values of a, if any, does limxa+f(x) equal a finite number? b. For what values of a, if any, does limxa+f(x)=? c. For what values of a, if any, does limxa+f(x)=?Steep secant lines a. Given the graph of f in the following figures, find the slope of the secant line that passes through (0, 0) and (h, f(h)) in terms of h, for h 0 and h 0. b. Analyze the limit of the slope of the secant line found in part (a) as h 0+ and h 0. What does this tell you about the line tangent to the curve at (0, 0)? 54. f(x) = x1/3Steep secant lines a. Given the graph of f in the following figures, find the slope of the secant line that passes through (0, 0) and (h, f(h)) in terms of h, for h 0 and h 0. b. Analyze the limit of the slope of the secant line found in part (a) as h 0+ and h 0. What does this tell you about the line tangent to the curve at (0, 0)? 55. f(x) = x2/3Evaluate x/(x+1) for x = 10,100, and 1000. What is limxxx+1?Describe the behavior of p(x)=3x3 as x and as xUse Theorem 2.7 to find the vertical and horizontal asymptotes of y=10x3x1.How do the functions e10x and e10x behave as x and as x?Explain the meaning of limxf(x)=10.Evaluate limxf(x) and limxf(x) using the figure.Infinite limits at infinity Determine the following limits. 15. limxx124EInfinite limits at infinity Determine the following limits. 17. limxx66EInfinite limits at infinity Determine the following limits. 23. limx(12x5)Infinite limits at infinity Determine the following limits. 22. limx2x8Limits at infinity Evaluate the following limits. 9. limx(3+10x2)Limits at infinity Evaluate the following limits. 10. limx(5+1x+10x2)Determine limxf(x)g(x) if f(x) 100,000 and g(x) as x .Limits at infinity Evaluate the following limits. 12. limx3+2x+4x2x2Evaluate limxex,limxex, and limxex.Describe the end behavior of g(x) = e2x.Suppose the function g satisfies the inequality 31x2g(x)3+1x2, for all nonzero values of x. Evaluate limxg(x) and limxg(x).The graph of g has a vertical asymptote at x = 2 and horizontal asymptotes at y = 1 and y = 3 (see figure). Determine the following limits: limxg(x), limxg(x), limx2g(x), and limx2+g(x)Limits at infinity Evaluate the following limits. 11. limcos2Limits at infinity Determine the following limits. 18.limt5t2+tsintt2Limits at infinity Evaluate the following limits. 13. limxcosx5xLimits at infinity Evaluate the following limits. 14. limx(5+100x+sin4x3x2)Infinite limits at infinity Determine the following limits. 19. limx(3x129x7)Infinite limits at infinity Determine the following limits. 20. limx(3x7+x2)Infinite limits at infinity Determine the following limits. 21. limx(3x16+2)Infinite limits at infinity Determine the following limits. 24. limx(2x8+4x3)Limits at infinity Determine the following limits. 25.limx14x3+3x22x21x3+x2+2x+1Limits at infinity Determine the following limits. 26.limx9x3+x253x4+4x2Limits at infinity Determine the following limits. 27.limx3x2+3xx+1Limits at infinity Determine the following limits. 28.limxx4+7x5+x2xLimits at infinity Determine the following limits. 29.limw15w2+3w+19w4+w3Limits at infinity Determine the following limits. 30.limx40x4+x2+5x64x8+x6Limits at infinity Determine the following limits. 31.limx16x2+xxLimits at infinity Determine the following limits. 32.limx6x24x2+16x4+x2Limits at infinity Determine the following limits. 33.limx(x2x4+3x2)(Hint: Multiply by x2+x4+3x2x2+x4+3x2 first.)Limits at infinity Determine the following limits. 34.limx(x+x25x)Limits at infinity Determine the following limits. 35limxsinxexLimits at infinity Determine the following limits. 36.limx(excosx+3)Rational functions Determine limxf(x) and limxf(x) for the following rational functions. Then give the horizontal asymptote of f (if any). 25. f(x)=4x20x+1Rational functions Determine limxf(x) and limxf(x) for the following rational functions. Then give the horizontal asymptote of f (if any). 26. f(x)=3x27x2+5xRational functions Determine limxf(x) and limxf(x) for the following rational functions. Then give the horizontal asymptote of f (if any). 27. f(x)=6x29x+83x2+2Horizontal asymptotes Determine limxf(x) and limxf(x) for the following functions Then give the horizontal asymptotes of f (if any). 40.f(x)=12x833x82x7Rational functions Determine limxf(x) and limxf(x) for the following rational functions. Then give the horizontal asymptote of f (if any). 29. f(x)=3x27x4+5x2Rational functions Determine limxf(x) and limxf(x) for the following rational functions. Then give the horizontal asymptote of f (if any). 31. f(x)=2x+13x42Rational functions Determine limxf(x) and limxf(x) for the following rational functions. Then give the horizontal asymptote of f (if any). 33. f(x)=40x5x216x42xHorizontal asymptotes Determine limxf(x) and limxf(x) for the following functions Then give the horizontal asymptotes of f (if any). 44.f(x)=6x2+14x4+3x+1Horizontal asymptotes Determine limxf(x) and limxf(x) for the following functions Then give the horizontal asymptotes of f (if any). 45.f(x)=12x44x89x4Horizontal asymptotes Determine limxf(x) and limxf(x) for the following functions Then give the horizontal asymptotes of f (if any). 46.f(x)=x2+12x+1Algebraic functions Determine limxf(x) and limxf(x) for the following functions. Then give the horizontal asymptote(s) of f (if any). 41. f(x)=4x3+12x3+16x6+148EAlgebraic functions Determine limxf(x) and limxf(x) for the following functions. Then give the horizontal asymptote(s) of f (if any). 43. f(x)=x6+834x2+3x4+1Algebraic functions Determine limxf(x) and limxf(x) for the following functions. Then give the horizontal asymptote(s) of f (if any). 44. f(x)=4x(3x9x2+1)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. c. Graph f and all 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. f(x)=x23x+6Slant (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. c. Graph f and all its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. 36. f(x)=x21x+2Slant (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. c. Graph f and all its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. 37. f(x)=x22x+53x2Slant (oblique) asymptotes Complete the following steps for the given functions a. Find the slant asymptote of f. b. Find the vertical asymptotes of f (if any). c. 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 54.f(x)=5x245x5Slant (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. c. Graph f and all its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. 39. f(x)=4x3+4x2+7x+41+x2Slant (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. c. Graph f and all its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph. 40. f(x)=3x22x+53x+4Transcendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 45. f(x) = 3exTranscendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 46. f(x) = 2xTranscendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 47. f(x) = 1 ln xTranscendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 48. f(x) = |ln x|Transcendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 49. f(x) = sin x.Transcendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 50. f(x)=50e2xExplain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The graph of a function can never cross one of its horizontal asymptotes.
A rational function f has both (where L is finite) and .
The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.
Steady states If a function f represents a system that varies in time, the existence of limtf(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady slate exists and give the steady -state value. 70. The population of a bacteria culture is given by p(t)2500t+1.Steady states If a function f represents a system that varies in time, the existence of limtf(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady slate exists and give the steady -state value. 71. The population of a culture of tumor cells is given by p(t)3500tt+1.Steady states If a function f represents a system that varies in time, the existence of limtf(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady slate exists and give the steady -state value. 72. The amount of drug (in milligrams) in the blood after an IV tube is inserted is m(t) = 200(1 2t).Steady states If a function f represents a system that varies in time, the existence of limtf(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady slate exists and give the steady -state value. 73. The value of an investment in dollars is given by v(t) = 1000e0.065t.Steady states If a function f represents a system that varies in time, the existence of limtf(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady slate exists and give the steady -state value. 74. The population of a colony of squirrels is given by p(t)=15003+2e0.1t.Steady states If a function f represents a system that varies in time, the existence of limtf(t) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady slate exists and give the steady -state value. 75. The amplitude of an oscillator is given by a(t)=2(t+sintt).Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 52. f(x)=x24x+3x1Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 53. f(x)=2x3+10x2+12xx3+2x2Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 54. f(x)=16x4+64x2+x22x24Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 55. f(x)=3x4+3x336x2x425x2+144Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then Identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a analyze limxaf(x) and limxa+f(x). 74. f(x)=x2(4x216x4+1)Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 57. f(x)=x29x(x3)Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then Identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a analyze limxaf(x) and limxa+f(x). 76.f(x)=x41x21Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 59. f(x)x2+2x+63x1Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 60. f(x)=1x2x(x+1)Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 61. f(x)=xx1Horizontal and vertical asymptotes a. Analyze limxf(x) and limxf(x), and then Identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote x = a, analyze limxaf(x) and limxa+f(x). 80. f(x)=3ex+10exAsymptotes Find the vertical and horizontal asymptotes of f(x)=cosx+2xx.End behavior for transcendental functions 82.Consider the graph of y=cot1x (see Section 1.4) and determine the following limits using the graph a.limxcot1x b.limxcot1xConsider the graph of y = sec1 x (see Section 1.4) and evaluate the following limits using the graph. Assume the domain is {x: |x| 1}. a. limxsec1x b. limxsec1xEnd behavior for transcendental functions 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+ex2. a. Determine its end behavior by analyzing limxcoshx and limxcoshx. b. Evaluate cosh 0. Use symmetry and part (a) to sketch a plausible graph for y = cosh x.End behavior for transcendental functions 65. The hyperbolic sine function is defined as sinhx=exex2. a. Determine its end behavior by analyzing limxsinhx and limxsinhx. b. Evaluate sinh 0. Use symmetry and pan (a) to sketch a plausible graph for y = sinh x.Sketching graphs Sketch a possible graph of a function f that satisfies all the given conditions. Be sure to identify all vertical and horizontal asymptotes. 66. f(1)=2,f(1)=2,f(0)=0,limxf(x)=1,limxf(x)=1Sketching graphs Sketch a possible graph of a function f that satisfies all the given conditions. Be sure to identify all vertical and horizontal asymptotes. 67. limx0+f(x)=,limx0f(x)=,limxf(x)=1,limxf(x)=288ELooking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6, 8, } is specified by the function f(n) = 2n, where n = 1, 2, 3, . The limit of such a sequence is limnf(n), 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. 77. {0,12,23,34,}, which is defined by f(n)=n1n, for n = 1, 2, 3,90E91EEnd behavior of a rational function Suppose f(x)=p(x)q(x) is a rational function, where p(x)=amxm+am1xm1++a2x2+a1x+a0, q(x)=bnxn+bn1xn1++b2x2+b1x+b0, am 0, and bn 0. a. Prove that if m = n, then limxf(x)=ambm. b. Prove that if m n, then limxf(x)=0.Horizontal and slant asymptotes a. Is it possible for a rational function to have both slant and horizontal asymptotes? Explain. b. Is it possible for an algebraic function to have two distinct slant asymptotes? Explain or give an example.End behavior of exponentials Use the following Instructions to determine the end behavior of f(x)=4ex+2e2x8ex+e2x. a. Evaluate limxf(x) by first dividing the numerator and denominator by e2x. b. Evaluate limxf(x) by first dividing the numerator and denominator by ex. c. Give the horizontal asymptote(s). d. Graph f to confirm your work in parts (a)(c).Find the horizontal asymptotes of each function using limits at infinity. 95.f(x)=2ex+3ex+1Find the horizontal asymptotes of each function using limits at infinity. 96.f(x)3e5x+7e6x9e5x+14e6xUse analytical methods to identify all the asymptotes of f(x)=lnx6lnx31 Plot a graph of the function with a graphing utility and then sketch a graph by hand, correcting any errors In the computer-generated graph.For what values of t in (0, 60) does the graph of y = c(t) in Figure 2.46b have a discontinuity?Modify the graphs of the functions t and g in Figure 2.50 to obtain functions that are continuous on [a, b].On what interval is f(x)=x1/4 continuous? On what interval is f(x)=x2/5 continuous?Show that f(x)=lnx4 is right-continuous at x = 1.Does the equation f(x)=x3+x+1=0 have a solution on the interval [1,1]? Explain.Which of the following functions are continuous for all values in their domain? Justify your answers. a. a(t) = altitude of a skydiver t seconds after jumping from a plane b. n(t) = number of quarters needed to park legally in a metered parking space for t minutes c. T(t) = temperature t minutes after midnight in Chicago on January 1 d. p(t) = number of points scored by a basketball player after t minutes of a basketball gameGive the three conditions that must be satisfied by a function to be continuous at a point.