home *** CD-ROM | disk | FTP | other *** search
/ NeXT Education Software Sampler 1992 Fall / NeXT Education Software Sampler 1992 Fall.iso / Mathematics / Notebooks / CSOMinesCalculus / Chapter2 / ans2.tex < prev    next >
LaTeX Document  |  1992-06-26  |  3.3 KB

open in: MacOS 8.1     |     Win98     |     DOS

browse contents    |     view JSON data     |     view as text


This file was processed as: LaTeX Document (document/latex).

ConfidenceProgramDetectionMatch TypeSupport
100% dexvert LaTeX Document (document/latex) magic Supported
1% dexvert Corel 10 Texture (image/corel10Texture) ext Unsupported
1% dexvert Croteam texture file (image/croteamTextureFile) ext Unsupported
1% dexvert Text File (text/txt) fallback Supported
100% file LaTeX document text default
99% file LaTeX document, ASCII text default
100% checkBytes Printable ASCII default
100% perlTextCheck Likely Text (Perl) default
100% siegfried fmt/281 LaTeX (Subdocument) default
100% detectItEasy Format: plain text[LF] default (weak)



hex view
+--------+-------------------------+-------------------------+--------+--------+
|00000000| 5c 64 6f 63 75 6d 65 6e | 74 73 74 79 6c 65 5b 31 |\documen|tstyle[1|
|00000010| 31 70 74 2c 66 6c 65 71 | 6e 2c 65 70 73 66 2c 63 |1pt,fleq|n,epsf,c|
|00000020| 61 6c 63 5d 7b 61 72 74 | 69 63 6c 65 7d 0a 5c 6d |alc]{art|icle}.\m|
|00000030| 61 72 6b 72 69 67 68 74 | 7b 43 68 61 70 74 65 72 |arkright|{Chapter|
|00000040| 20 32 3a 20 41 6e 73 77 | 65 72 73 20 32 7d 0a 5c | 2: Answ|ers 2}.\|
|00000050| 62 65 67 69 6e 7b 64 6f | 63 75 6d 65 6e 74 7d 0a |begin{do|cument}.|
|00000060| 0a 5c 42 66 7b 43 68 61 | 70 74 65 72 20 32 3a 20 |.\Bf{Cha|pter 2: |
|00000070| 41 6e 73 77 65 72 73 20 | 32 20 5c 68 66 69 6c 6c |Answers |2 \hfill|
|00000080| 20 4a 61 63 6b 20 4b 2e | 20 43 6f 68 65 6e 20 20 | Jack K.| Cohen |
|00000090| 5c 68 66 69 6c 6c 20 43 | 6f 6c 6f 72 61 64 6f 20 |\hfill C|olorado |
|000000a0| 53 63 68 6f 6f 6c 20 6f | 66 20 4d 69 6e 65 73 7d |School o|f Mines}|
|000000b0| 0a 5c 62 65 67 69 6e 7b | 65 6e 75 6d 65 72 61 74 |.\begin{|enumerat|
|000000c0| 65 7d 0a 5c 69 74 65 6d | 0a 5c 62 65 67 69 6e 7b |e}.\item|.\begin{|
|000000d0| 76 65 72 62 61 74 69 6d | 7d 0a 66 5b 78 5f 5d 20 |verbatim|}.f[x_] |
|000000e0| 3a 3d 20 20 78 5e 33 20 | 2b 20 33 78 5e 32 0a 66 |:= x^3 |+ 3x^2.f|
|000000f0| 5b 32 5d 0a 20 20 20 20 | 32 30 0a 20 20 20 20 0a |[2]. |20. .|
|00000100| 66 5b 78 5d 20 2f 2e 20 | 78 20 2d 3e 20 32 0a 20 |f[x] /. |x -> 2. |
|00000110| 20 20 20 32 30 0a 20 20 | 20 20 0a 66 5b 74 5d 0a | 20. | .f[t].|
|00000120| 20 20 20 20 20 20 20 32 | 20 20 20 20 33 0a 20 20 | 2| 3. |
|00000130| 20 20 33 20 74 20 20 2b | 20 74 0a 20 20 20 20 0a | 3 t +| t. .|
|00000140| 20 66 5b 63 61 74 5d 0a | 20 20 20 20 20 20 20 20 | f[cat].| |
|00000150| 20 32 20 20 20 20 20 20 | 33 0a 20 20 20 20 33 20 | 2 |3. 3 |
|00000160| 63 61 74 20 20 2b 20 63 | 61 74 0a 20 20 20 20 0a |cat + c|at. .|
|00000170| 54 61 62 6c 65 5b 66 5b | 6e 5d 2c 20 7b 6e 2c 20 |Table[f[|n], {n, |
|00000180| 31 2c 20 35 7d 5d 0a 20 | 20 20 20 7b 34 2c 20 32 |1, 5}]. | {4, 2|
|00000190| 30 2c 20 35 34 2c 20 31 | 31 32 2c 20 32 30 30 7d |0, 54, 1|12, 200}|
|000001a0| 0a 20 20 20 20 0a 50 6c | 6f 74 5b 66 5b 78 5d 2c |. .Pl|ot[f[x],|
|000001b0| 20 7b 78 2c 20 2d 32 2c | 20 32 7d 5d 0a 20 20 20 | {x, -2,| 2}]. |
|000001c0| 20 53 65 65 20 46 69 67 | 75 72 65 20 31 2e 0a 5c | See Fig|ure 1..\|
|000001d0| 65 6e 64 7b 76 65 72 62 | 61 74 69 6d 7d 0a 0a 5c |end{verb|atim}..\|
|000001e0| 62 65 67 69 6e 7b 66 69 | 67 75 72 65 7d 5b 68 74 |begin{fi|gure}[ht|
|000001f0| 62 5d 0a 5c 65 70 73 66 | 79 73 69 7a 65 20 39 30 |b].\epsf|ysize 90|
|00000200| 70 74 0a 5c 63 65 6e 74 | 65 72 6c 69 6e 65 7b 5c |pt.\cent|erline{\|
|00000210| 65 70 73 66 66 69 6c 65 | 7b 61 6e 73 32 70 31 2e |epsffile|{ans2p1.|
|00000220| 65 70 73 7d 7d 0a 5c 63 | 61 70 74 69 6f 6e 7b 50 |eps}}.\c|aption{P|
|00000230| 6c 6f 74 20 6f 66 20 24 | 78 5e 33 20 2b 20 33 78 |lot of $|x^3 + 3x|
|00000240| 5e 32 24 2e 7d 20 0a 5c | 65 6e 64 7b 66 69 67 75 |^2$.} .\|end{figu|
|00000250| 72 65 7d 0a 5c 69 74 65 | 6d 20 20 24 5c 6c 69 6d |re}.\ite|m $\lim|
|00000260| 5f 7b 78 20 5c 72 69 67 | 68 74 61 72 72 6f 77 20 |_{x \rig|htarrow |
|00000270| 30 7d 24 20 7b 5c 4c 61 | 72 67 65 20 24 5c 66 72 |0}$ {\La|rge $\fr|
|00000280| 61 63 7b 5c 73 71 72 74 | 7b 78 20 2b 20 32 35 7d |ac{\sqrt|{x + 25}|
|00000290| 20 2d 20 35 7d 7b 78 7d | 24 7d 20 69 73 20 31 2f | - 5}{x}|$} is 1/|
|000002a0| 31 30 2e 0a 09 5c 62 65 | 67 69 6e 7b 65 6e 75 6d |10...\be|gin{enum|
|000002b0| 65 72 61 74 65 7d 0a 09 | 5c 69 74 65 6d 20 46 69 |erate}..|\item Fi|
|000002c0| 67 75 72 65 20 33 20 20 | 73 68 6f 77 73 20 74 68 |gure 3 |shows th|
|000002d0| 65 20 6f 75 74 70 75 74 | 20 66 72 6f 6d 20 0a 09 |e output| from ..|
|000002e0| 0a 09 5c 54 74 7b 50 6c | 6f 74 5b 28 53 71 72 74 |..\Tt{Pl|ot[(Sqrt|
|000002f0| 5b 78 20 2b 20 32 35 5d | 20 2d 20 35 29 2f 78 2c |[x + 25]| - 5)/x,|
|00000300| 20 5c 7b 78 2c 20 2d 32 | 2c 20 32 5c 7d 2c 20 50 | \{x, -2|, 2\}, P|
|00000310| 6c 6f 74 52 61 6e 67 65 | 20 2d 3e 20 5c 7b 30 2c |lotRange| -> \{0,|
|00000320| 20 2e 35 5c 7d 5d 7d 0a | 0a 5c 62 65 67 69 6e 7b | .5\}]}.|.\begin{|
|00000330| 66 69 67 75 72 65 7d 5b | 68 74 62 5d 0a 5c 65 70 |figure}[|htb].\ep|
|00000340| 73 66 79 73 69 7a 65 20 | 31 31 30 70 74 0a 5c 63 |sfysize |110pt.\c|
|00000350| 65 6e 74 65 72 6c 69 6e | 65 7b 5c 65 70 73 66 66 |enterlin|e{\epsff|
|00000360| 69 6c 65 7b 61 6e 73 32 | 70 32 2e 65 70 73 7d 7d |ile{ans2|p2.eps}}|
|00000370| 0a 5c 63 61 70 74 69 6f | 6e 7b 50 6c 6f 74 20 66 |.\captio|n{Plot f|
|00000380| 6f 72 20 70 72 6f 62 6c | 65 6d 20 32 2e 7d 20 0a |or probl|em 2.} .|
|00000390| 5c 65 6e 64 7b 66 69 67 | 75 72 65 7d 0a 0a 09 5c |\end{fig|ure}...\|
|000003a0| 69 74 65 6d 20 48 65 72 | 65 20 69 73 20 74 68 65 |item Her|e is the|
|000003b0| 20 6f 75 74 70 75 74 20 | 66 72 6f 6d 0a 09 0a 09 | output |from....|
|000003c0| 5c 54 74 7b 54 61 62 6c | 65 5b 5c 7b 78 2c 20 28 |\Tt{Tabl|e[\{x, (|
|000003d0| 53 71 72 74 5b 78 20 2b | 20 32 35 5d 20 2d 20 35 |Sqrt[x +| 25] - 5|
|000003e0| 29 2f 78 5c 7d 2c 20 5c | 7b 78 2c 20 2d 31 2f 35 |)/x\}, \|{x, -1/5|
|000003f0| 2c 20 31 2f 35 2c 20 31 | 2f 32 30 5c 7d 5d 20 2f |, 1/5, 1|/20\}] /|
|00000400| 2f 4e 20 2f 2f 54 61 62 | 6c 65 46 6f 72 6d 7d 0a |/N //Tab|leForm}.|
|00000410| 5c 62 65 67 69 6e 7b 76 | 65 72 62 61 74 69 6d 7d |\begin{v|erbatim}|
|00000420| 0a 20 20 20 20 2d 30 2e | 32 20 20 20 20 30 2e 31 |. -0.|2 0.1|
|00000430| 30 30 32 30 31 0a 20 20 | 20 20 2d 30 2e 31 35 20 |00201. | -0.15 |
|00000440| 20 20 30 2e 31 30 30 31 | 35 0a 20 20 20 20 2d 30 | 0.1001|5. -0|
|00000450| 2e 31 20 20 20 20 30 2e | 31 30 30 31 0a 20 20 20 |.1 0.|1001. |
|00000460| 20 2d 30 2e 30 35 20 20 | 20 30 2e 31 30 30 30 35 | -0.05 | 0.10005|
|00000470| 0a 20 20 20 20 30 20 20 | 20 20 20 20 20 49 6e 64 |. 0 | Ind|
|00000480| 65 74 65 72 6d 69 6e 61 | 74 65 0a 20 20 20 20 30 |etermina|te. 0|
|00000490| 2e 30 35 20 20 20 20 30 | 2e 30 39 39 39 35 0a 20 |.05 0|.09995. |
|000004a0| 20 20 20 30 2e 31 20 20 | 20 20 20 30 2e 30 39 39 | 0.1 | 0.099|
|000004b0| 39 30 30 32 0a 20 20 20 | 20 30 2e 31 35 20 20 20 |9002. | 0.15 |
|000004c0| 20 30 2e 30 39 39 38 35 | 30 34 0a 20 20 20 20 30 | 0.09985|04. 0|
|000004d0| 2e 32 20 20 20 20 20 30 | 2e 30 39 39 38 30 30 38 |.2 0|.0998008|
|000004e0| 0a 5c 65 6e 64 7b 76 65 | 72 62 61 74 69 6d 7d 0a |.\end{ve|rbatim}.|
|000004f0| 49 74 20 6d 61 79 20 73 | 65 65 6d 20 6d 6f 72 65 |It may s|eem more|
|00000500| 20 6e 61 74 75 72 61 6c | 20 74 6f 20 60 60 66 6f | natural| to ``fo|
|00000510| 72 63 65 27 27 20 5c 4d | 6d 61 5c 20 74 6f 20 75 |rce'' \M|ma\ to u|
|00000520| 73 65 20 64 65 63 69 6d | 61 6c 20 61 72 69 74 68 |se decim|al arith|
|00000530| 6d 65 74 69 63 20 64 75 | 72 69 6e 67 20 74 68 65 |metic du|ring the|
|00000540| 20 63 61 6c 63 75 6c 61 | 74 69 6f 6e 20 69 6e 73 | calcula|tion ins|
|00000550| 74 65 61 64 20 6f 66 20 | 65 76 6f 6b 69 6e 67 20 |tead of |evoking |
|00000560| 5c 54 74 7b 4e 7d 20 61 | 66 74 65 72 77 61 72 64 |\Tt{N} a|fterward|
|00000570| 73 2c 20 70 65 72 68 61 | 70 73 20 6c 69 6b 65 20 |s, perha|ps like |
|00000580| 74 68 69 73 3a 0a 0a 20 | 5c 54 74 7b 54 61 62 6c |this:.. |\Tt{Tabl|
|00000590| 65 5b 5c 7b 78 2c 20 28 | 53 71 72 74 5b 78 20 2b |e[\{x, (|Sqrt[x +|
|000005a0| 20 32 35 5d 20 2d 20 35 | 29 2f 78 5c 7d 2c 20 5c | 25] - 5|)/x\}, \|
|000005b0| 7b 78 2c 20 2d 2e 32 2c | 20 2e 32 2c 20 2e 30 35 |{x, -.2,| .2, .05|
|000005c0| 5c 7d 5d 20 2f 2f 54 61 | 62 6c 65 46 6f 72 6d 7d |\}] //Ta|bleForm}|
|000005d0| 2e 0a 20 0a 48 6f 77 65 | 76 65 72 2c 20 77 69 74 |.. .Howe|ver, wit|
|000005e0| 68 20 69 6e 2d 65 78 61 | 63 74 20 61 72 69 74 68 |h in-exa|ct arith|
|000005f0| 6d 65 74 69 63 2c 20 74 | 68 65 20 75 73 65 72 20 |metic, t|he user |
|00000600| 68 61 73 20 74 6f 20 62 | 65 20 61 6c 65 72 74 20 |has to b|e alert |
|00000610| 65 6e 6f 75 67 68 20 74 | 6f 20 72 65 6a 65 63 74 |enough t|o reject|
|00000620| 20 6f 75 74 70 75 74 20 | 74 68 61 74 20 69 73 20 | output |that is |
|00000630| 5c 45 6d 7b 74 6f 6f 7d | 20 63 6c 6f 73 65 20 74 |\Em{too}| close t|
|00000640| 6f 20 74 68 65 20 73 69 | 6e 67 75 6c 61 72 20 70 |o the si|ngular p|
|00000650| 6f 69 6e 74 3a 0a 5c 62 | 65 67 69 6e 7b 76 65 72 |oint:.\b|egin{ver|
|00000660| 62 61 74 69 6d 7d 0a 2d | 30 2e 32 20 20 20 20 20 |batim}.-|0.2 |
|00000670| 20 20 20 20 20 20 20 20 | 30 2e 31 30 30 32 30 31 | |0.100201|
|00000680| 0a 2d 30 2e 31 35 20 20 | 20 20 20 20 20 20 20 20 |.-0.15 | |
|00000690| 20 20 30 2e 31 30 30 31 | 35 0a 2d 30 2e 31 20 20 | 0.1001|5.-0.1 |
|000006a0| 20 20 20 20 20 20 20 20 | 20 20 20 30 2e 31 30 30 | | 0.100|
|000006b0| 31 0a 2d 30 2e 30 35 20 | 20 20 20 20 20 20 20 20 |1.-0.05 | |
|000006c0| 20 20 20 30 2e 31 30 30 | 30 35 0a 20 20 20 20 20 | 0.100|05. |
|000006d0| 20 20 20 20 20 20 2d 31 | 37 0a 2d 31 2e 33 38 37 | -1|7.-1.387|
|000006e0| 37 38 20 31 30 20 20 20 | 20 20 20 30 2e 0a 30 2e |78 10 | 0..0.|
|000006f0| 30 35 20 20 20 20 20 20 | 20 20 20 20 20 20 20 30 |05 | 0|
|00000700| 2e 30 39 39 39 35 0a 30 | 2e 31 20 20 20 20 20 20 |.09995.0|.1 |
|00000710| 20 20 20 20 20 20 20 20 | 30 2e 30 39 39 39 30 30 | |0.099900|
|00000720| 32 0a 30 2e 31 35 20 20 | 20 20 20 20 20 20 20 20 |2.0.15 | |
|00000730| 20 20 20 30 2e 30 39 39 | 38 35 30 34 0a 30 2e 32 | 0.099|8504.0.2|
|00000740| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 30 2e | | 0.|
|00000750| 30 39 39 38 30 30 38 0a | 5c 65 6e 64 7b 76 65 72 |0998008.|\end{ver|
|00000760| 62 61 74 69 6d 7d 0a 0a | 09 5c 69 74 65 6d 20 49 |batim}..|.\item I|
|00000770| 66 20 45 78 61 6d 70 6c | 65 20 31 32 20 73 65 65 |f Exampl|e 12 see|
|00000780| 6d 73 20 6f 62 73 63 75 | 72 65 2c 20 61 73 6b 20 |ms obscu|re, ask |
|00000790| 79 6f 75 72 20 69 6e 73 | 74 72 75 63 74 6f 72 20 |your ins|tructor |
|000007a0| 66 6f 72 20 68 65 6c 70 | 2e 0a 09 5c 69 74 65 6d |for help|...\item|
|000007b0| 20 20 41 73 20 69 6e 64 | 69 63 61 74 65 64 2c 20 | As ind|icated, |
|000007c0| 24 5c 73 71 72 74 7b 78 | 20 2b 20 32 35 7d 20 20 |$\sqrt{x| + 25} |
|000007d0| 5c 61 70 70 72 6f 78 20 | 35 20 2b 20 78 2f 31 30 |\approx |5 + x/10|
|000007e0| 24 2c 20 73 6f 20 20 77 | 65 20 63 61 6e 20 72 65 |$, so w|e can re|
|000007f0| 70 6c 61 63 65 20 74 68 | 65 20 67 69 76 65 6e 20 |place th|e given |
|00000800| 6c 69 6d 69 74 20 62 79 | 20 24 5c 6c 69 6d 5f 7b |limit by| $\lim_{|
|00000810| 78 20 5c 72 69 67 68 74 | 61 72 72 6f 77 20 30 7d |x \right|arrow 0}|
|00000820| 24 20 7b 5c 4c 61 72 67 | 65 20 24 5c 66 72 61 63 |$ {\Larg|e $\frac|
|00000830| 7b 28 35 20 2b 20 78 2f | 31 30 29 20 2d 35 7d 7b |{(5 + x/|10) -5}{|
|00000840| 78 7d 24 7d 20 24 20 3d | 20 31 2f 31 30 24 2e 0a |x}$} $ =| 1/10$..|
|00000850| 09 5c 69 74 65 6d 20 52 | 75 6e 6e 69 6e 67 20 74 |.\item R|unning t|
|00000860| 68 65 20 69 6e 64 69 63 | 61 74 65 64 20 63 6f 64 |he indic|ated cod|
|00000870| 65 20 67 69 76 65 73 3a | 0a 5c 62 65 67 69 6e 7b |e gives:|.\begin{|
|00000880| 76 65 72 62 61 74 69 6d | 7d 0a 30 2e 31 20 20 20 |verbatim|}.0.1 |
|00000890| 20 20 20 20 30 2e 30 39 | 39 39 30 30 32 0a 30 2e | 0.09|99002.0.|
|000008a0| 30 31 20 20 20 20 20 20 | 30 2e 30 39 39 39 39 0a |01 |0.09999.|
|000008b0| 30 2e 30 30 31 20 20 20 | 20 20 30 2e 30 39 39 39 |0.001 | 0.0999|
|000008c0| 39 39 0a 30 2e 30 30 30 | 31 20 20 20 20 30 2e 30 |99.0.000|1 0.0|
|000008d0| 39 39 39 39 39 39 0a 30 | 2e 30 30 30 30 31 20 20 |999999.0|.00001 |
|000008e0| 20 30 2e 31 0a 5c 65 6e | 64 7b 76 65 72 62 61 74 | 0.1.\en|d{verbat|
|000008f0| 69 6d 7d 0a 57 69 74 68 | 20 5c 54 74 7b 78 20 2d |im}.With| \Tt{x -|
|00000900| 3e 20 61 20 2d 20 31 2f | 31 30 5c 5e 7b 7d 6b 7d |> a - 1/|10\^{}k}|
|00000910| 2c 20 77 65 20 67 65 74 | 3a 0a 5c 62 65 67 69 6e |, we get|:.\begin|
|00000920| 7b 76 65 72 62 61 74 69 | 6d 7d 0a 2d 30 2e 31 20 |{verbati|m}.-0.1 |
|00000930| 20 20 20 20 20 20 30 2e | 31 30 30 31 0a 2d 30 2e | 0.|1001.-0.|
|00000940| 30 31 20 20 20 20 20 20 | 30 2e 31 30 30 30 31 0a |01 |0.10001.|
|00000950| 2d 30 2e 30 30 31 20 20 | 20 20 20 30 2e 31 30 30 |-0.001 | 0.100|
|00000960| 30 30 31 0a 2d 30 2e 30 | 30 30 31 20 20 20 20 30 |001.-0.0|001 0|
|00000970| 2e 31 0a 2d 30 2e 30 30 | 30 30 31 20 20 20 30 2e |.1.-0.00|001 0.|
|00000980| 31 0a 5c 65 6e 64 7b 76 | 65 72 62 61 74 69 6d 7d |1.\end{v|erbatim}|
|00000990| 0a 0a 5c 6e 6f 69 6e 64 | 65 6e 74 20 4e 6f 74 65 |..\noind|ent Note|
|000009a0| 3a 20 57 65 20 68 61 76 | 65 20 73 75 70 70 6c 69 |: We hav|e suppli|
|000009b0| 65 64 20 61 20 66 75 6e | 63 74 69 6f 6e 20 63 61 |ed a fun|ction ca|
|000009c0| 6c 6c 65 64 20 5c 54 74 | 7b 4c 69 6d 69 74 54 61 |lled \Tt|{LimitTa|
|000009d0| 62 6c 65 7d 20 74 68 61 | 74 20 75 73 65 73 20 74 |ble} tha|t uses t|
|000009e0| 68 69 73 20 63 6f 64 65 | 20 61 73 20 69 74 73 20 |his code| as its |
|000009f0| 63 6f 72 65 2e 0a 0a 09 | 5c 69 74 65 6d 20 59 6f |core....|\item Yo|
|00000a00| 75 72 20 6f 70 69 6e 69 | 6f 6e 20 69 73 20 6d 6f |ur opini|on is mo|
|00000a10| 72 65 20 69 6d 70 6f 72 | 74 61 6e 74 20 74 68 61 |re impor|tant tha|
|00000a20| 6e 20 6d 69 6e 65 2e 0a | 09 5c 65 6e 64 7b 65 6e |n mine..|.\end{en|
|00000a30| 75 6d 65 72 61 74 65 7d | 0a 0a 5c 69 74 65 6d 20 |umerate}|..\item |
|00000a40| 28 32 2e 32 2e 38 29 20 | 24 5c 6c 69 6d 5f 7b 74 |(2.2.8) |$\lim_{t|
|00000a50| 20 5c 72 69 67 68 74 61 | 72 72 6f 77 20 32 7d 24 | \righta|rrow 2}$|
|00000a60| 20 7b 5c 4c 61 72 67 65 | 20 24 5c 66 72 61 63 7b | {\Large| $\frac{|
|00000a70| 74 5e 32 20 2b 20 32 74 | 20 2d 20 35 7d 7b 74 5e |t^2 + 2t| - 5}{t^|
|00000a80| 33 20 2d 20 32 74 7d 24 | 7d 20 24 20 3d 20 33 2f |3 - 2t}$|} $ = 3/|
|00000a90| 34 24 2e 0a 20 0a 5c 69 | 74 65 6d 20 28 32 2e 32 |4$.. .\i|tem (2.2|
|00000aa0| 2e 32 32 29 20 24 5c 6c | 69 6d 5f 7b 78 20 5c 72 |.22) $\l|im_{x \r|
|00000ab0| 69 67 68 74 61 72 72 6f | 77 20 39 7d 24 20 7b 5c |ightarro|w 9}$ {\|
|00000ac0| 4c 61 72 67 65 20 24 5c | 66 72 61 63 7b 33 20 2d |Large $\|frac{3 -|
|00000ad0| 20 5c 73 71 72 74 7b 78 | 7d 7d 7b 39 20 2d 20 78 | \sqrt{x|}}{9 - x|
|00000ae0| 7d 24 7d 20 24 20 3d 20 | 31 2f 36 24 2e 0a 0a 5c |}$} $ = |1/6$...\|
|00000af0| 69 74 65 6d 20 28 32 2e | 32 2e 33 32 29 20 24 5c |item (2.|2.32) $\|
|00000b00| 6c 69 6d 5f 7b 78 20 5c | 72 69 67 68 74 61 72 72 |lim_{x \|rightarr|
|00000b10| 6f 77 20 31 7d 20 24 20 | 7b 5c 4c 61 72 67 65 20 |ow 1} $ |{\Large |
|00000b20| 24 5c 66 72 61 63 7b 78 | 5e 34 20 2d 20 31 7d 7b |$\frac{x|^4 - 1}{|
|00000b30| 78 20 2d 20 31 7d 24 7d | 20 24 20 3d 20 34 24 2e |x - 1}$}| $ = 4$.|
|00000b40| 0a 0a 5c 69 74 65 6d 20 | 28 32 2e 32 2e 33 35 29 |..\item |(2.2.35)|
|00000b50| 20 24 5c 6c 69 6d 5f 7b | 78 20 5c 72 69 67 68 74 | $\lim_{|x \right|
|00000b60| 61 72 72 6f 77 20 30 7d | 24 20 7b 5c 4c 61 72 67 |arrow 0}|$ {\Larg|
|00000b70| 65 20 24 5c 66 72 61 63 | 7b 31 7d 7b 78 7d 20 5c |e $\frac|{1}{x} \|
|00000b80| 6c 65 66 74 28 20 5c 66 | 72 61 63 7b 31 7d 7b 78 |left( \f|rac{1}{x|
|00000b90| 20 2b 20 35 7d 20 2d 20 | 5c 66 72 61 63 7b 31 7d | + 5} - |\frac{1}|
|00000ba0| 7b 35 7d 20 5c 72 69 67 | 68 74 29 24 7d 20 24 20 |{5} \rig|ht)$} $ |
|00000bb0| 3d 20 2d 31 2f 31 30 24 | 2e 0a 0a 5c 69 74 65 6d |= -1/10$|...\item|
|00000bc0| 20 28 32 2e 32 2e 33 36 | 29 20 24 5c 6c 69 6d 5f | (2.2.36|) $\lim_|
|00000bd0| 7b 78 20 5c 72 69 67 68 | 74 61 72 72 6f 77 20 38 |{x \righ|tarrow 8|
|00000be0| 7d 24 20 7b 5c 4c 61 72 | 67 65 20 24 5c 66 72 61 |}$ {\Lar|ge $\fra|
|00000bf0| 63 7b 78 5e 7b 32 2f 33 | 7d 20 2d 20 34 7d 7b 78 |c{x^{2/3|} - 4}{x|
|00000c00| 20 2d 20 38 7d 24 7d 20 | 24 20 3d 20 31 2f 33 24 | - 8}$} |$ = 1/3$|
|00000c10| 2e 0a 20 20 20 20 20 20 | 20 20 20 54 6f 20 64 6f |.. | To do|
|00000c20| 20 74 68 69 73 20 61 6c | 67 65 62 72 61 69 63 61 | this al|gebraica|
|00000c30| 6c 6c 79 20 69 73 20 61 | 20 62 69 74 20 74 72 69 |lly is a| bit tri|
|00000c40| 63 6b 65 79 2e 20 20 46 | 61 63 74 6f 72 20 74 68 |ckey. F|actor th|
|00000c50| 65 20 6e 75 6d 65 72 61 | 74 6f 72 3a 20 20 24 28 |e numera|tor: $(|
|00000c60| 78 5e 7b 31 2f 33 7d 20 | 20 2d 20 32 29 28 78 5e |x^{1/3} | - 2)(x^|
|00000c70| 7b 31 2f 33 7d 20 20 2b | 20 32 29 24 0a 09 20 61 |{1/3} +| 2)$.. a|
|00000c80| 6e 64 20 74 68 65 20 64 | 65 6e 6f 6d 69 6e 61 74 |nd the d|enominat|
|00000c90| 6f 72 3a 20 24 28 78 5e | 7b 31 2f 33 7d 20 20 2d |or: $(x^|{1/3} -|
|00000ca0| 20 32 29 28 78 5e 7b 32 | 2f 33 7d 20 2b 20 32 20 | 2)(x^{2|/3} + 2 |
|00000cb0| 78 5e 7b 31 2f 33 7d 20 | 2b 20 34 29 24 2e 20 20 |x^{1/3} |+ 4)$. |
|00000cc0| 54 68 65 6e 20 63 61 6e | 63 65 6c 20 74 68 65 20 |Then can|cel the |
|00000cd0| 63 6f 6d 6d 6f 6e 20 0a | 09 20 74 65 72 6d 20 61 |common .|. term a|
|00000ce0| 6e 64 20 73 75 62 73 74 | 69 74 75 74 65 20 24 20 |nd subst|itute $ |
|00000cf0| 78 20 3d 20 38 24 2e 0a | 5c 65 6e 64 7b 65 6e 75 |x = 8$..|\end{enu|
|00000d00| 6d 65 72 61 74 65 7d 0a | 5c 65 6e 64 7b 64 6f 63 |merate}.|\end{doc|
|00000d10| 75 6d 65 6e 74 7d 0a | |ument}. | |
+--------+-------------------------+-------------------------+--------+--------+