2.4 Infinite Limits. An Overview Infinite Limits Finding Infinite Limits Analytically Quick Quiz SECTION 2.4 EXERCISES
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1 Section 2.4 Infinite Limits 2.4 Infinite Limits Two more limit scenarios are frequently encountered in calculus and are discussed in this and the following sections. An infinite limit occurs when function values increase or decrease without bound near a point. The other type of limit, known as a limit at infinity, occurs when the independent variable x increases or decreases without bound. The ideas behind infinite limits and limits at infinity are quite different. Therefore, it is important to distinguish these limits and the methods used to calculate them. An Overview Infinite Limits Finding Infinite Limits Analytically Quick Quiz SECTION 2.4 EXERCISES Review Questions. Use a graph to explain the meaning of lim xøa + fhxl=-. 2. Use a graph to explain the meaning of lim xøa fhxl=. 3. What is a vertical asymptote? 4. Consider the function FHxL= fhxl with ghal= 0. Does F necessarily have a vertical asymptote at x= a? Explain ghxl your reasoning. fhxl 5. Suppose fhxlø00 and ghxlø0 with ghxl< 0 as xø 2. Determine lim xø2 ghxl. 6. Evaluate lim and lim xø3 - x-3 xø3 + x-3. Basic Skills T 7. Analyzing infinite limits numerically Compute the values of fhxl= x+ them to determine lim xø fhxl. in the following table and use Hx- L2 x x+ Hx- L x x+ Hx- L 2 Copyright 204 Pearson Education, Inc.
2 2 Chapter 2 Limits 8. Analyzing infinite limits graphically Use the graph of f HxL = lim fhxl. xø3 x to determine lim Ix x-3m xø- fhxl and 9. Analyzing infinite limits graphically The graph of f in the figure has vertical asymptotes at x= and x=2. Find the following limits, if possible. fhxl xø - fhxl xø + fhxl xø d. lim fhxl xø2 - e. lim fhxl xø2 + f. lim fhxl xø2 0. Analyzing infinite limits graphically The graph of g in the figure has vertical asymptotes at x= 2 and x=4. Analyze the following limits. ghxl xø2 - ghxl xø2 + ghxl xø2 d. lim ghxl xø4 - e. lim ghxl xø4 + f. lim ghxl xø4 Copyright 204 Pearson Education, Inc.
3 Section 2.4 Infinite Limits 3. Analyzing infinite limits graphically The graph of h in the figure has vertical asymptotes at x=-2 and x=3. Investigate the following limits. hhxl xø-2 - hhxl xø-2 + hhxl xø-2 d. lim hhxl xø3 - e. lim hhxl xø3 + f. lim hhxl xø3 2. Analyzing infinite limits graphically The graph of p in the figure has vertical asymptotes at x=-2 and x=3. Investigate the following limits. phxl xø-2 - phxl xø-2 + phxl xø-2 d. lim phxl xø3 - e. lim phxl xø3 + f. lim phxl xø3 Copyright 204 Pearson Education, Inc.
4 4 Chapter 2 Limits T 3. Analyzing infinite limits graphically Graph the function f HxL = using a graphing utility with the window x 2 - 2Dµ@-0, 0D. Use the graph to determine the following limits. fhxl xø0 - fhxl xø0 + fhxl xø - d. lim fhxl xø + e -x T 4. Analyzing infinite limits graphically Graph the function f HxL = using a graphing utility. (Experiment xhx+ 2L2 with your choice of a graphing window.) Use the graph to determine the following limits. xø-2 + fhxl xø-2 fhxl xø0 - fhxl d. lim xø0 + fhxl 5. Sketching graphs Sketch a possible graph of a function f, together with vertical asymptotes, satisfying all of the following conditions. f HL=0 f H3L is undefined lim xø3 f HxL= lim f HxL=- xø0 + lim f HxL= xø2 lim f HxL= xø4-6. Sketching graphs Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all of the following conditions. gh2l= gh5l=- lim xø4 ghxl=- lim ghxl= xø7 - lim ghxl=- xø Evaluating limits analytically Evaluate the following limits or state that they do not exist. 7. x- 2 xø2 + b. lim xø2 - x- 2 Copyright 204 Pearson Education, Inc.
5 Section 2.4 Infinite Limits 5 xø2 x-2 8. xø3 + 2 Hx- 3L 3 xø3-2 Hx- 3L 3 xø3 2 Hx-3L 3 9. xø4 + x-5 Hx- 4L 2 xø4 - x-5 Hx- 4L 2 xø4 x-5 Hx-4L xø + x-2 Hx- L 3 xø - x-2 Hx- L 3 xø x-2 Hx-L 3 2. Hx- LHx-2L xø3 + Hx-3L Hx- LHx-2L xø3 - Hx-3L Hx-LHx-2L xø3 Hx- 3L 22. Hx-4L xø-2 + xhx+ 2L Hx-4L xø-2 - xhx+ 2L xø-2 Hx- 4L xhx+2l 23. Copyright 204 Pearson Education, Inc.
6 6 Chapter 2 Limits 24. xø2 + x2-4 x+ 3 Hx-2L 2 xø2 - x2-4 x+ 3 Hx-2L 2 xø2 x2-4 x+3 Hx- 2L 2 xø-2 + x3-5 x2 + 6 x x 4-4 x 2 xø-2 - x3-5 x2 + 6 x x 4-4 x 2 x 3-5 x x xø-2 x 4-4 x 2 x 3-5 x x xø2 x 4-4 x lim xø0 x3-5 x2 x lim tø5 4 t2-00 t lim xø + x2-5 x+6 x- 28. lim zø4 z-5 Iz 2-0 z+ 24M Location of vertical asymptotes Analyze the following limits and find the vertical asymptotes of fhxl= x- 5. x 2-25 xø5 fhxl xø-5 - fhxl xø-5 + fhxl 30. Location of vertical asymptotes Analyze the following limits and find the vertical asymptotes of x+ 7 fhxl=. x 4-49 x 2 fhxl xø7 - fhxl xø7 + Copyright 204 Pearson Education, Inc.
7 Section 2.4 Infinite Limits 7 T xø-7 fhxl d. lim fhxl xø Finding vertical asymptotes Find all vertical asymptotes x=a of the following functions. For each value of a, evaluate lim fhxl, lim fhxl, and lim fhxl. xøa + xøa- xøa 3. fhxl= x2-9 x fhxl= cos x x 2-5 x+6 x x x+ 33. fhxl= x 3-4 x x 34. fhxl= x3-0 x x x 2-8 x Trigonometric limits Determine the following limits. 35. lim qø0 + cscq 36. lim xø0 - csc x 37. lim cot xl xø0 +H lim qøpê2 + 3 tanq 39. Analyzing infinite limits graphically Graph the function y=tan x with the pdµ@-0, 0D. Use the graph to analyze the following limits. xøpê2 + tan x xøpê2 - tan x xø-pê2 + tan x d. lim xø-pê2 - tan x T 40. Analyzing infinite limits graphically Graph the function y=sec x tan x with the pdµ@-0, 0D. Use the graph to analyze the following limits. sec x tan x xøpê2 + sec x tan x xøpê2- sec x tan x xø-pê2 + d. lim sec x tan x xø-pê2- Copyright 204 Pearson Education, Inc.
8 8 Chapter 2 Limits Further Explorations 4. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The line x= is a vertical asymptote of the function fhxl= x2-7 x+6. x 2 - b. The line x=- is a vertical asymptote of the function fhxl= x2-7 x+6. x 2 - c. If g has a vertical asymptote at x = and lim ghxl=, then lim ghxl=. xø + xø Finding a function with vertical asymptotes Find polynomials p and q such that p q is undefined at x= and x=2, but p q has a vertical asymptote only at x= 2. Sketch a graph of your function. 43. Finding a function with infinite limits Give a formula for a function f that satisfies lim fhxl= and xø6 + lim fhxl=-. xø6-44. Matching Match functions a-f with graphs A-F in the figure without using a graphing utility. x a. fhxl= x 2 + x b. fhxl= x 2 - c. fhxl= x 2 - x d. fhxl= Hx- L 2 e. fhxl= Hx- L 2 f. fhxl= x x+ Copyright 204 Pearson Education, Inc.
9 Section 2.4 Infinite Limits 9 T Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. 45. fhxl= x2-3 x+2 x 0 - x ghxl=2-ln x 2 e x 47. hhxl = Hx+L phxl=sec p x, for x < ghql=tan pq 0 Copyright 204 Pearson Education, Inc.
10 0 Chapter 2 Limits p 50. qhsl = s- sin s 5. fhxl= x sec x 52. ghxl=e êx Additional Exercises 53. Limits with a parameter Let fhxl= x2-7 x+2. x-a a. For what values of a, if any, does lim fhxl equal a finite number? xøa + b. For what values of a, if any, does lim fhxl=? xøa + c. For what values of a, if any, does lim fhxl=-? xøa Steep secant lines a. Given the graph of f in the following figures, find the slope of the secant line that passes through H0, 0L and Hh, fhhll 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 hø 0 + and hø0 -. What does this tell you about the tangent line to the curve at H0, 0L? 54. fhxl= x ê3 55. fhxl= x 2ê3 Copyright 204 Pearson Education, Inc.
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