(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e8, 24, "B Univers 65 Bold"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e6, 18, "Garamond"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, e6, 14, "I Garamond LightItalic"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, a20, 18, "B Univers 65 Bold"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, a15, 14, "AGaramond Semibold"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, a12, 12, "Garamond"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "AGaramond"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "AGaramond"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M7, 12, "Garamond"; fontset = leftheader, inactive, L2, 12, "Garamond"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M7, 12, "Garamond"; fontset = leftfooter, inactive, L2, 12, "Garamond"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, N55, 12, "Garamond"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, 12, "Times"; showRuler; currentKernel; ] :[font = text; inactive; preserveAspect; leftWrapOffset = 17; leftNameWrapOffset = 1] Mathematics 162 Laboratory 6 Week of March 1, 1993 Name: _____________________________ Lab Partner: ___________________________ Consulted with: ____________________________________________________________ :[font = smalltext; inactive; preserveAspect; right] © Lafayette College, 1994. :[font = title; inactive; preserveAspect] Limits and Indeterminate Forms :[font = text; inactive; preserveAspect] In this laboratory you will use numerical methods to examine limits involving indeterminate forms. :[font = section; inactive; preserveAspect] Part 0: A Demonstration :[font = text; inactive; preserveAspect] Consider the function :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 149; pictureWidth = 82; pictureHeight = 26] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.317073 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.317073 scale 1 string 82 26 1 [82 0 0 26 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FC7FFFFFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1C7FFFFFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC70001FFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE38E3FFFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE07FFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE38FFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FFFFFFFFFFF FC FFFFFFFC7FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FF1FFFFFFFFFFF FC FF8FFFE3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFE00FFFFFFFFFFFF FC FFF1FFE3FFC7FC71FFFC0007FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFF1FF1FFFF8E3FE3FFFFFFFFF8000000000000000000000000000000000 00 FFFE3F1FFFFFE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFFE3F1FFFFF1C7E3FFC0007FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFFFC71FFFFF1FF03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFFE001FFFFF1FF03FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFFFF8E3FFFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFFFFF03FFFFFFF1FFFFFFFFFFFFF8FFFFFFFFFFF803FE00FFF007FFF1FF 1C FFFFFFFC7FFFFF8FFFFFFFFFFFFFF8FFFFFFFFFFC7FC71FF1F8FF8FFFE3F E0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFC7FFF1FF1FFFF8FFFFC7 1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFE001FFFC7FFF1FF1FF007FFFFFF 1C FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFC7FFF1FF1F8FFFFFFFF8 E0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFC7FC71FF1F8FF8FFFFF8 FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFF803FE00FFF007FFFFF8 FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] and the problem of computing its limit as x goes to 0. If x is very close to 0, then both the numerator and denominator of f (x) are close to 0 and it is not clear whether or not the values of fÊ(x) are close to any one particular number. This limit involves the indeterminate form 0/0. ;[s] 15:0,0;42,1;43,0;59,1;60,0;124,1;126,0;127,1;128,0;194,1;196,0;197,1;198,0;265,1;283,0;289,-1; 2:8,16,12,Chicago,0,12,0,0,0;7,16,12,Chicago,2,12,0,0,0; :[font = text; inactive; preserveAspect] We can use a graph of f to get a good idea of what the limit is. From ;[s] 3:0,0;22,1;23,0;71,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] f[x_] := (1 - Cos[x])/x^2 :[font = input; preserveAspect] Plot[f[x], {x, -1, 1}] :[font = text; inactive; preserveAspect] we can see pretty clearly that the limit is close to 1/2. We can try to get more evidence to support our guess by examining a table of values of f. The command ;[s] 3:0,0;146,1;147,0;162,-1; 2:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] Table[{t = 0.1^n, f[t]}, {n, 1, 10}] :[font = text; inactive; preserveAspect] prints a table of approximate values of f (0.1), f (0.01), f (0.001), ..., f (0.0000000001). The first few of these values seem to confirm our guess that the limit is 1/2, but something strange happens when we compute f (0.0000001), f (0.00000001), and the others. Round-off error has crept in and given us very poor accuracy in our computations. We can overcome this by asking Mathematica to do its arithmetic with more precision. The following command asks for the same table of function values, but tells Mathematica to use 50 digits in doing its computations. ;[s] 17:0,0;40,1;42,0;49,1;51,0;59,1;61,0;75,1;77,0;219,1;221,0;234,1;236,0;381,1;392,0;512,1;523,0;568,-1; 2:9,16,12,Chicago,0,12,0,0,0;8,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] Table[{t = 1/10^n, N[f[t],50]}, {n, 1, 10}] :[font = text; inactive; preserveAspect] This table seems to confirm our guess that :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 149; pictureWidth = 84; pictureHeight = 38] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.452381 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.452381 scale 1 string 84 38 1 [84 0 0 38 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFE3FE3FFFFFFFFFFFC01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 0000 FFFFFFFC7FC7FFFFFF1FFE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFF0 FFFFFFFF8E3FFFFFFFE3FE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 1FF0 FFFFFFFFFE3FFFFFFFFC7E3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF E3F0 FFFFFFFFF1C7FC0000000E3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FC70 FFFFFFFFF1FF1FFFFFFC7E3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF80 FFFFFFFFF1FF1FFFFFE3FE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7 FF80 FFFFFFFFFFFFFFFFFF1FFFC01FFFFFFFFFFFFFF1FFFFFE3FFFFFFFFFFFF8 FF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFF8FF8FF8FC7FFFFFFFFFFFF 0070 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FF8FFF1FF1C7FFF0001FFFFF FFF0 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFFC7FC7FFFE38FF8FFFFFFFFFE00 0000 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFFF8FC7FFFFF8FF8FFFFFFFFFFFF FFF0 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFFF8FC7FFFFC71F8FFF0001FFFFF E3F0 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFFFF1C7FFFFC7FC0FFFFFFFFFFFF E3F0 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFFF8007FFFFC7FC0FFFFFFFFFFFF E3F0 FFFFFFFFFFC7E381F81F8FFFFFFFFFFFFFFFE38FFFFFFFC7FFFFFFFFFFFF E3F0 FFFFFFFFFFC7038E07E07FFFFFFFFFFFFFFFFC0FFFFFFFC7FFFFFFFFFFFF E3F0 FFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFFFFFF1FFFFFE3FFFFFFFFFFFFF E3F0 FFFFFFFFFE07E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF E3F0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 03F0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF E3F0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFF0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] In the following parts of this lab, you should try to get your answers using the first type of table. But beware of strange output and consider using higher-precision arithmetic if it is needed. :[font = section; inactive; preserveAspect] Part 1: Limits as x Approaches 0 ;[s] 3:0,0;19,1;20,0;34,-1; 2:2,23,18,Chicago,0,18,0,0,0;1,23,18,Chicago,2,18,0,0,0; :[font = text; inactive; preserveAspect] In each of the following, write the indeterminate form which arises in the problem and estimate the indicated limit. Your estimates should be accurate to three places after the decimal. :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 131; pictureHeight = 38] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.290076 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.290076 scale 1 string 131 38 1 [131 0 0 38 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FF1FFFFFFFFFFFE00FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE3FFFFFF8FFF1FF1FFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC71FFFFFFFF1FF1FF1FFFFFFFFFFF FFFFFFFFFFC7FC7FFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FFFFFFFFE3F1FF1FFFFFFFFFFF FFFFFFFFFFF8FF8FFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E3FE000000071FF1FFFFFFFFFFF FFFFFFFFFFFF1C7FFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFE3F1FF1FFFFFFFFFFF FFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFF1FF1FF1FFFFFFFFFFF FFFFFFFFFFFFE38FFFFFFFFFFFFFFFFFFFFFFF80 FF8FFFFC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFE00FFFFFFFFFFFF FFFFFFFFFFFFE3FE3FFFFFFFFFFFFFFFFFFFFF80 FC7FC0038FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFE3FE3FFFFFFFFFFFFFFFFFFFFF80 FC7E3FE38FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0007FF80 E3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFE00000 000000000000000000000000000007FFFFFFFF80 E3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 E3FFC003F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0007FF80 E3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 E3FFC01FF1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FC7FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFE3F1C0FC0FC7FFFFFFFFFFFFFFF FFFFC7FFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFF80 FC7FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFE381C703F03FFFFFFFFFFFF8E3F E3FE3FFF8FFFFFFFFFFFF1FF1F8FFFFFFFFFFF80 FF8FFFFC7FFFFFFFFFFFFFFFFFFFFFFFFE3FFFFFFFFFFFFFFFFFFFFF8E3F E3FE3FFF8FFFFFFFFFFFFE3FE38FFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03F1FFFFFFFFFFFFFFFFFFF8E3F E3F1FFFF8FFFFF8FFFFFFFC71FF1FFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E3F E3F1FFFF8FFFFF8FFFFFFFFF1FF1FFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E3F E3F1FFFF8FFFE0003FFFFFF8E3F1FFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E07 E3F1FFFF8FFFFF8FFFFFFFF8FF81FFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E38 1FF1FFFF8FFFFF8FFFFFFFF8FF81FFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFF FFFE3FFC0FFFFFFFFFFFFFFFFF8FFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFF FFFE3FFF8FFFFFFFFFFFFFFFFF8FFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFC7FFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF80 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] The indeterminate form in this limit is: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 136; pictureHeight = 38] %! 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Since we are now interested in values of f (x) for large values of x, we need to modify the table command a bit. For example, in part a), use the command: ;[s] 7:0,0;153,1;155,0;156,1;157,0;179,1;180,0;268,-1; 2:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = input; preserveAspect] Table[{x = 10.0^n, x^50/E^x}, {n, 1, 10}] :[font = text; inactive; preserveAspect] Be sure to notice that x is given as a decimal number (10.0). This indicates to Mathematica that numerical answers (rather than exact answers, in terms of e) are called for. ;[s] 7:0,0;23,1;24,0;81,1;92,0;156,1;157,0;175,-1; 2:4,16,12,Chicago,0,12,0,0,0;3,16,12,Chicago,2,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 103; pictureHeight = 38] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.368932 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.368932 scale 1 string 103 38 1 [103 0 0 38 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FC7FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFE3FFFFFFFFFFFFFFFFFC0 1FFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1C7FFFFFFFC7FFF0381FFFFFFFFFC7 FC7FFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFF8FF8FC7E3FFFFFFFFC7 FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE38FF80000001FF0381FFFFFFFFFF8 000FC7FFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE3FFFFFF8FFFFFFFFFFFFFFFFF8 FF8E3FFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE3FFFFFC7FFFFFFFFFFFFFFFFFF 1FF03FFFFFFFFFFFF8 FF8FFFFC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFFFFFFFFFFFFFFFFFF E001C7FFFFFFFFFFF8 FC7FC0038FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FF8FF8FFFFFFFFFFF8 FC7E3FE38FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFC0007FF8 E3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFE00000 000000007FFFFFFFF8 E3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 E3FFC003F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFF8FF8 FFFFFFFFFFC0007FF8 E3FE3FE3F1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFF1FF 1FFFFFFFFFFFFFFFF8 E3FFC01FF1FFFFFFFFFFFFFFFFFFFFFFFE3F1C7FC7FC7FFFFFFFFFFFFE38 FFFFFFFFFFFFFFFFF8 FC7FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFE3F1C0FC0FC7FFFFFFFFFFFFFF8 FC01C01FFFFFFFFFF8 FC7FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFE381C703F03FFFFFFFFFFFFFFC7 03FE3FE3FFFFFFFFF8 FF8FFFFC7FFFFFFFFFFFFFFFFFFFFFFFFE3FFFFFFFFFFFFFFFFFFFFFFFC7 FC7E3FE3FFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03F1FFFFFFFFFFFFFFFFFFFFFC7 FC7E3FE3FFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF E0003FE3FFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF E3FE3FE3FFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF E000001FFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFF8 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; 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inactive; preserveAspect] The indeterminate form in this limit is: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 146; pictureHeight = 38] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.260274 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.260274 scale 1 string 146 38 1 [146 0 0 38 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FF1FFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FE3FFFFFF8FFFFC0FF81FFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC71FFFFFFFF1FFE3F1C7E3FFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FFFFFFFFE3FE3FE3FE3FFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8E3FE00000007E3F1C7E3FFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFE3FFC0FF81FFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFF1FFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FF8FFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFC7 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FC7FC003FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7 FFF1FF1FFFFFFFFFFFFFFFF8FFFFFFFFFFFF1FF1FFFFFFFFFC FC7E3FE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7 FFFE3FE3FFFFFFFFFFFFFFF8FFFFFFFFFFFFE3FE3FE0003FFC E3FE3FE3FFC7FFFFFFFFFFFFFFFFFFFFFFF8FC71FF1FF1FFFFFFFFFFFFC7 FFFFC71FFFFFFFF1FFFFFFF8FFFFFFFFFFFFFC71FFFFFFFFFC E3FE3FE3FFC7FFFFFFFFFFFFFFFFFFFFFFF8FC71FF1FF1FFFFFFFFFFFFC0 FFFFFF1C0007FFF1FFFFFFF8FFFFC003FFFFFFF1FFFFFFFFFC E3FE3FE3FFC7FFFFFFFFFFFFFFFFFFFFFFF8FC71FF1FF1FFFFFFFFFFFE38 FFFFF8E38FFFFC0007FFFFF8FFFFFFFFFFFFFF8E3FE0003FFC E3FE3FE3FFC7FFFFFFFFFFFFFFFFFFFFFFF8FC71FF1FF1FFFFFFFFFFFE38 FFFFF8FF81FFFFF1FFFFFFF8FFFFFFFFFFFFFF8FF8FFFFFFFC E3FFC003FFC7FFFFFFFFFFFFFFFFFFFFFFF8FC71FF1FF1FFFFFFFFFFF038 FFFFF8FF8E3FFFF1FFFFFFF8FFFFFFFFFFFFFF8FF8FFFFFFFC FC7FFFE3FE3FFFFFFFFFFFFFFFFFFFFFFFF8FC703F03F1FFFFFFFFFFFE38 FFFFFFFFFFC7FFFFFFFFFFC0FFFFFFFFFFFFFFFFFFFFFFFFFC FC7FFFE3FE3FFFFFFFFFFFFFFFFFFFFFFFF8E071C0FC0FFFFFFFFFFFFFFF 1FFFFFFC7FC7FFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFC FF8FFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFFFFFFFFFFFFFFFFF 1FFFFFFF803FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FC7FFFFFFFFFFFFFFFFFFFFF 000000000000000000000000007FFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] The indeterminate form in this limit is: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 0; pictureWidth = 146; pictureHeight = 38] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.260274 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.260274 scale 1 string 146 38 1 [146 0 0 38 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FF8FFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF1FF1FFFFFFC7FFFE07FC0FFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE38FFFFFFFF8FFF1F8E3F1FFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8FFFFFFFFF1FF1FF1FF1FFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC71FF00000003F1F8E3F1FFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FC7FFFFFF1FFE07FC0FFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FC7FFFFF8FFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FF8FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFE3F FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FC7FC01FF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3F FF8FF8FFFFFFFFFFFFFFF1FF1FFFFFFFFFFF1FF1FFFFFFFFFC FC7E3FE3F1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3F FFF1FF1FFFFFFFFFFFFFFE3FE3FFFFFFFFFFE3FE3FE0003FFC E3FE3FFFFE3FFFFFFFFFFFFFFFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE3F FFFE38FFFFFFFF8FFFFFFFC71FFFFFFFFFFFFC71FFFFFFFFFC E3FE0003FE3FFFFFFFFFFFFFFFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07 FFFFF8E0003FFF8FFFFFFFFF1FFFC003FFFFFFF1FFFFFFFFFC E3FE3FE3FE3FFFFFFFFFFFFFFFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFF1C7 FFFFC71C7FFFE0003FFFFFF8E3FFFFFFFFFFFF8E3FE0003FFC E3FE3FE3FE3FFFFFFFFFFFFFFFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFF1C7 FFFFC7FC0FFFFF8FFFFFFFF8FF8FFFFFFFFFFF8FF8FFFFFFFC E3FFC01FFE3FFFFFFFFFFFFFFFFFFFFFFFC7E38FF8FF8FFFFFFFFFFF81C7 FFFFC7FC71FFFF8FFFFFFFF8FF8FFFFFFFFFFF8FF8FFFFFFFC FC7FFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFC7E381F81F8FFFFFFFFFFFF1C7 FFFFFFFFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FC7FFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFC7038E07E07FFFFFFFFFFFFFF8 FFFFFFE3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FF8FFFFF8FFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFC01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07E3FFFFFFFFFFFFFFFFFFFFF8 000000000000000000000000007FFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] The indeterminate form in this limit is: :[font = text; inactive; preserveAspect] :[font = section; inactive; preserveAspect] Part 3: Limits of Integrals :[font = text; inactive; preserveAspect] Recall that if f is continuous and positive over the interval [a, b], then the area bounded by y = f (x), y = 0, x = a, and x = b is ;[s] 23:0,0;15,1;16,0;64,1;65,0;67,1;68,0;96,1;97,0;100,1;102,0;103,1;104,0;107,1;108,0;114,1;115,0;118,1;119,0;125,1;126,0;129,1;130,0;136,-1; 2:12,16,12,Chicago,0,12,0,0,0;11,16,12,Chicago,2,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 164; pictureWidth = 55; pictureHeight = 43] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.781818 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.781818 scale 1 string 55 43 1 [55 0 0 43 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 03FFC01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 007E3F1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FC0FC01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF8FFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF8FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFC7FFFFF8FFFFFFFFFFFFFF8 FF81FFFFFFFFFF8FFFE3FE3FE3F1FFFFF000FC7FC0 FF81FFFFFFFFFFF1FFE3FFC7FC71FFFFF1FF1F8FF8 FF81FFFFFFFFFFF1FF1FFFF8E3FE3FFFF1FF1FF1C0 FF81FFFFFFFFFFFE3F1FFFFFE3FE3FFFFE3FE3FFC0 FF81FFFFFFFFFFFE3F1FFFFF1C7E3FFFFE3FE3FE38 FF81FFFFFFFFFFFFC71FFFFF1FF03FFFFFC7FC7E38 FF81FFFFFFFFFFFE001FFFFF1FF03FFFFFF8007E38 FF81FFFFFFFFFFFFF8E3FFFFFFF1FFFFFFFFFF8FF8 FF81FFFFFFFFFFFFFF03FFFFFFF1FFFFFFFFFF8FF8 FF81FFFFFFFFFFFFFFFC7FFFFF8FFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFF1FFE00FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFF03FE3F1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFE00E3FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFC0FC7E3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFC01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] Mathematica will attempt to evaluate this integral (and find the described area) when we enter the command Integrate[f[x], {x, a, b}]. ;[s] 4:0,1;11,0;107,2;133,0;135,-1; 3:2,16,12,Chicago,0,12,0,0,0;1,16,12,Chicago,2,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] (a) Define the function f (x) = 1/lnx, and plot it over the interval [2, 8]. Set the range of the plot using the option PlotRange->{0,2}. Print out a copy of the graph, and shade the region bounded by y = f (x), y = 0, x = 2, and x = 5. Use Mathematica to find the area of this region, and write it below. ;[s] 23:0,0;24,1;26,0;27,1;28,0;36,1;37,0;121,2;137,0;203,1;204,0;207,1;209,0;210,1;211,0;214,1;215,0;221,1;222,0;232,1;233,0;244,1;255,0;309,-1; 3:12,16,12,Chicago,0,12,0,0,0;10,16,12,Chicago,2,12,0,0,0;1,13,10,Courier,1,12,0,0,0; :[font = text; inactive; preserveAspect] :[font = text; inactive; preserveAspect] (b) Create a table of values for the integral below for t = 10.0n, for n = 1 to 10. ;[s] 7:0,0;56,1;57,0;64,2;65,0;71,1;72,0;84,-1; 3:4,16,12,Chicago,0,12,0,0,0;2,16,12,Chicago,2,12,0,0,0;1,18,12,Chicago,34,9,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 165; pictureWidth = 54; pictureHeight = 43] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.796296 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.796296 scale 1 string 54 43 1 [54 0 0 43 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 03FE0003FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 007FC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FC0FF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF8FFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF8FFFE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF803FE3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81C01FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFE3FFFFFC7FFFFFFFFFFFFFC0 FF81FFFFFFFFFC7FFF1FF1FF1F8FFFFF8007E3FE00 FF81FFFFFFFFFF8FFF1FFE3FE38FFFFF8FF8FC7FC0 FF81FFFFFFFFFF8FF8FFFFC71FF1FFFF8FF8FF8E00 FF81FFFFFFFFFFF1F8FFFFFF1FF1FFFFF1FF1FFE00 FF81FFFFFFFFFFF1F8FFFFF8E3F1FFFFF1FF1FF1C0 FF81FFFFFFFFFFFE38FFFFF8FF81FFFFFE3FE3F1C0 FF81FFFFFFFFFFF000FFFFF8FF81FFFFFFC003F1C0 FF81FFFFFFFFFFFFC71FFFFFFF8FFFFFFFFFFC7FC0 FF81FFFFFFFFFFFFF81FFFFFFF8FFFFFFFFFFC7FC0 FF81FFFFFFFFFFFFFFE3FFFFFC7FFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FF81FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFF1FFFF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFF03FFC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFE00FC7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFC0FF8FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFC003FFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0 pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] Estimate the limit: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 144; pictureWidth = 96; pictureHeight = 44] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.458333 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.458333 scale 1 string 96 44 1 [96 0 0 44 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FF8000FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC01FF1FFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FE3FFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFC7FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFF8FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE00FF8FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07007FFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFE3FFFFFF8FFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFE3FFFFFFF1FFFFFE0703FFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFC7FFFFFFE3FFFF1F8FC7FFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFC7E00000007FFFE0703FFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFF8FFFFFFE3FFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFC01FFFFF1FFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFF1FFFF8FFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFF8FFFFFF1FF FFFFFFFFFFFF FFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFF1FFFC7FC7FC7E3F FFFE001F8FF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFE3FFC7FF8FF8E3F FFFE3FE3F1FF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFE3FE3FFFF1C7FC7 FFFE3FE3FE38 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFC7E3FFFFFC7FC7 FFFFC7FC7FF8 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFC7E3FFFFE38FC7 FFFFC7FC7FC7 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFF8E3FFFFE3FE07 FFFFF8FF8FC7 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFC003FFFFE3FE07 FFFFFF000FC7 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFFF1C7FFFFFFE3F FFFFFFFFF1FF FFFFFFFFFFC7E381F81F8FFFFFFFFFFFFE07FFFFFFFFFFFFE07FFFFFFE3F FFFFFFFFF1FF FFFFFFFFFFC7038E07E07FFFFFFFFFFFFE07FFFFFFFFFFFFFF8FFFFFF1FF FFFFFFFFFFFF FFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFE07E3FFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFE3FFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFF1FFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF803F1FFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FE3FFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000FFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFF pop grestore %% End of Graphics MathPictureEnd %% End of picture :[font = text; inactive; preserveAspect] (c) Repeat this technique to estimate the limit: :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 148; pictureWidth = 87; pictureHeight = 44] %! %%Creator: Mathematica MathPictureStart % Start of picture % Scaling calculations 0 1 0 1 [ [ 0.000000 0.000000 0 0 ] [ 1.000000 0.505747 0 0 ] ] MathScale % Start of Graphics 0 setgray 0 setlinewidth gsave 0.000000 0.000000 translate 1.000000 0.505747 scale 1 string 87 44 1 [87 0 0 44 0 0] { currentfile 1 index readhexstring pop } false 3 colorimage FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FF8000FFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC01FF1FFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FE3FFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFC7FFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE3FFF8FFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE00FF8FFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07007FFFFF8FF8FFFFFFFFFFFF FFFFF8 FFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFF1FF1FFFFFFFFFFF FFFFF8 FFFFFFE3FFFFFF8FFFFFFFFFFFFFFFFFFE07FFFFFFFFFE38FFFFFFFFFFFF FFFFF8 FFFFFFE3FFFFFFF1FFFFFE0703FFFFFFFE07FFFFFFFFFFF8FC01FFFFFFFF FFFFF8 FFFFFFFC7FFFFFFE3FFFF1F8FC7FFFFFFE07FFFFFFFFFFC703FE3FFFFFFF FFFFF8 FFFFFFFC7E00000007FFFE0703FFFFFFFE07FFFFFFFFFFC7FC7E3FFFFFFF FFFFF8 FFFFFFFF8FFFFFFE3FFFFFFFFFFFFFFFFE07FFFFFFFFFFC7FC01FFFFFFFF FFFFF8 FFFFFFFC01FFFFF1FFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFF8FFFFFFFFF FFFFF8 FFFFFFFFF1FFFF8FFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFF1FFFFFFFF FFFFF8 FFFFFFFFF1FFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFE0003FFFF000 FC7FC0 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFF1FF 1F8FF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFE00000000007FFF1FF 1FF1C0 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFE3F E3FFC0 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFE3F E3FE38 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFC7 FC7E38 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFF8 007E38 FFFFFFFFFFC7E38FF8FF8FFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFFF FF8FF8 FFFFFFFFFFC7E381F81F8FFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFFF FF8FF8 FFFFFFFFFFC7038E07E07FFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFFF FFFFF8 FFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFFF FFFFF8 FFFFFFFFFE07E3FFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFF81FFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFF1FFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFE07FFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFE3FFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC0FFF1FFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF803F1FFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF03FE3FFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000FFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFC7FFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFF8 pop grestore %% End of Graphics MathPictureEnd %% End of picture ^*)