Problem 1QC: Which of the following limits exist?
Problem 2QC: Give an example of a set that contains none of its boundary points.
Problem 3QC: Can the limit be evaluated by direct substitution?
Problem 4QC: What is the analog of the Two-Path Test for functions of a single variable?
Problem 5QC Problem 1E Problem 2E: Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths in... Problem 3E: What does it mean to say that limits of polynomials may be evaluated by direct substitution? Problem 4E: Suppose (a, b) is on the boundary of the domain of f. Explain how you would determine whether... Problem 5E: Explain how examining limits along multiple paths may prove the nonexistence of a limit. Problem 6E: Explain why evaluating a limit along a finite number of paths does not prove the existence of a... Problem 7E: What three conditions must be met for a function f to be continuous at the point (a, b)? Problem 8E: Let R be the unit disk {(x, y): x2 + y2 1} with (0, 0) removed. Is (0, 0) a boundary point of R? Is... Problem 9E: At what points of 2 is a rational function of two variables continuous? Problem 10E Problem 11E: Evaluate lim(x,y)(5,5)x2y2x+y Problem 12E: Let f(x)=x22xy2+1x22xy2+1 Use the Two-Path Test to show that lim(x,y)(1,0)f(x) does not exist.... Problem 13E: Limits of functions Evaluate the following limits. 11.lim(x,y)(2,9)101 Problem 14E: Limits of functions Evaluate the following limits. 12.lim(x,y)(1,3)(3x+4y2) Problem 15E: Limits of functions Evaluate the following limits. 13.lim(x,y)(3,3)(4x2y2) Problem 16E: Limits of functions Evaluate the following limits. 14.lim(x,y)(2,1)(xy83x2y3) Problem 17E: Limits of functions Evaluate the following limits. 15.lim(x,y)(0,)cosxy+sinxy2y Problem 18E: Limits of functions Evaluate the following limits. 16.lim(x,y)(e2,4)lnxy Problem 19E: Limits of functions Evaluate the following limits. 17.lim(x,y)(2,0)x23xy2x+y Problem 20E: Limits of functions Evaluate the following limits. 18.lim(u,v)(1,1)10uv2v2u2+v2 Problem 21E Problem 22E: Limits at boundary points Evaluate the following limits. 20.lim(x,y)(1,2)y2+2xyy+2x Problem 23E: Limits at boundary points Evaluate the following limits. 21.lim(x,y)(3,1)x27xy+12y2x3y Problem 24E: Limits at boundary points Evaluate the following limits. 22.lim(x,y)(1,1)2x2xy3y2x+y Problem 25E: Limits at boundary points Evaluate the following limits. 23.lim(x,y)(2,2)y24xy2x Problem 26E Problem 27E: Limits at boundary points Evaluate the following limits. 25.lim(x,y)(1,2)yx+1yx1 Problem 28E Problem 29E Problem 30E Problem 31E: Nonexistence of limits Use the Two-Path Test to prove that the following limits do not exist.... Problem 32E Problem 33E: Nonexistence of limits Use the Two-Path Test to prove that the following limits do not exist.... Problem 34E: Nonexistence of limits Use the Two-Path Test to prove that the following limits do not exist.... Problem 35E: Continuity At what points of 2 are the following functions continuous? 33.f(x,y)=x2+2xyy3 Problem 36E: Continuity At what points of 2 are the following functions continuous? 34.f(x,y)=xyx2y2+1 Problem 37E: Continuity At what points of 2 are the following functions continuous? 35.p(x,y)=4x2y2x4+y2 Problem 38E: Continuity At what points of 2 are the following functions continuous? 36.S(x,y)=2xyx2y2 Problem 39E: Continuity At what points of 2 are the following functions continuous? 37.f(x,y)=2x(y2+1) Problem 40E: Continuity At what points of 2 are the following functions continuous? 38.f(x,y)=x2+y2x(y21) Problem 41E: Continuity At what points of 2 are the following functions continuous?... Problem 42E: Continuity At what points of 2 are the following functions continuous?... Problem 43E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 44E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 45E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 46E: Continuity of composite functions At what points of 2 are the following functions continuous? 44.g... Problem 47E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 48E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 49E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 50E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 51E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 52E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 53E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 54E: Continuity of composite functions At what points of 2 are the following functions continuous?... Problem 55E: Limits of functions of three variables Evaluate the following limits. 53.lim(x,y,z)(1,ln2,3)zexy Problem 56E: Limits of functions of three variables Evaluate the following limits. 54.lim(x,y,z)(0,1,0)(1+y)lnexz Problem 57E: Limits of functions of three variables Evaluate the following limits.... Problem 58E: Limits of functions of three variables Evaluate the following limits.... Problem 59E: Limits of functions of three variables Evaluate the following limits.... Problem 60E: Limits of functions of three variables Evaluate the following limits.... Problem 61E Problem 62E: Miscellaneous limits Use the method of your choice to evaluate the following limits.... Problem 63E: Miscellaneous limits Use the method of your choice to evaluate the following limits.... Problem 64E: Miscellaneous limits Use the method of your choice to evaluate the following limits.... Problem 65E Problem 66E: Miscellaneous limits Use the method of your choice to evaluate the following limits.... Problem 67E: Miscellaneous limits Use the method of your choice to evaluate the following limits.... Problem 68E Problem 69E: Miscellaneous limits Use the method of your choice to evaluate the following limits.... Problem 70E Problem 71E: Limits of composite functions Evaluate the following limits. 77.lim(x,y)(1,0)sinxyxy Problem 72E: Limits of composite functions Evaluate the following limits. 80.lim(x,y)(0,/2)1cosxy4x2y3 Problem 73E: Limits of composite functions Evaluate the following limits. 79.lim(x,y)(0,2)(2xy)xy Problem 74E Problem 75E Problem 76E Problem 77E: Piecewise function Let f(x,y)={sin(x2+y21)x2+y21ifx2+y21bifx2+y2=1. Find the value of b for which f... Problem 78E Problem 79E Problem 80E Problem 81E Problem 82E Problem 83E: Nonexistence of limits Show that lim(x,y)(0,0)axmynbxm+n+cym+n does not exist when a, b, and c are... Problem 84E Problem 85E Problem 86E: Limit proof Use the formal definition of a limit to prove that lim(x,y)(a,b)y=b. (Hint: Take = .) Problem 87E: Limit proof Use the formal definition of a limit to prove thatlim(x,y)(a,b)(x+y)=a+b. (Hint: Take... Problem 88E: Proof of Limit Law 1 Use the formal definition of a limit to prove... Problem 89E: Proof of Limit Law 3 Use the formal definition of a limit to prove... format_list_bulleted
Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage