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<P>
;SPMlt;P;SPMgt;
Chapter 2: Answers 2 Jack K. Cohen  Colorado School of Mines

<P>
;SPMlt;OL;SPMgt;
;SPMlt;LI;SPMgt;;SPMlt;PRE;SPMgt;
f[x_] :=  x^3 + 3x^2
f[2]
    20

<P>
f[x] /. x -;SPMamp;gt; 2
    20

<P>
f[t]
       2    3
    3 t  + t

<P>
f[cat]
         2      3
    3 cat  + cat

<P>
Table[f[n], <#1#>n, 1, 5<#1#>]
    <#2#>4, 20, 54, 112, 200<#2#>

<P>
Plot[f[x], <#3#>x, -2, 2<#3#>]
    See Figure 1.
;SPMlt;/PRE;SPMgt;

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;SPMlt;P;SPMgt;

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;SPMlt;DIV class=;SPMquot;CENTER;SPMquot;;SPMgt;;SPMlt;A ID=;SPMquot;10;SPMquot;;SPMgt;;SPMlt;/A;SPMgt;
;SPMlt;TABLE;SPMgt;
;SPMlt;CAPTION class=;SPMquot;BOTTOM;SPMquot;;SPMgt;;SPMlt;STRONG;SPMgt;Figure:;SPMlt;/STRONG;SPMgt;
Plot of ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math1#<I>x</I><SUP>3</SUP> +3<I>x</I><SUP>2</SUP>
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;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;3;SPMlt;/SUP;SPMgt; +3;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;.;SPMlt;/CAPTION;SPMgt;
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<tex2html_image_mark>#figure4#
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;SPMlt;/TABLE;SPMgt;
;SPMlt;/DIV;SPMgt;

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;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math2#<tex2html_image_mark>#tex2html_wrap_inline281#

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;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math4#<tex2html_image_mark>#tex2html_wrap_inline285#

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;SPMlt;IMG
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;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; is 1/10.

<P>
;SPMlt;OL;SPMgt;
;SPMlt;LI;SPMgt;Figure 3  shows the output from 

<P>
;SPMlt;P;SPMgt;
Plot[(Sqrt[x + 25] - 5)/x, <#14#>x, -2, 2<#14#>, PlotRange -;SPMamp;gt; <#15#>0, .5<#15#>]

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;SPMlt;P;SPMgt;

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;SPMlt;DIV class=;SPMquot;CENTER;SPMquot;;SPMgt;;SPMlt;A ID=;SPMquot;19;SPMquot;;SPMgt;;SPMlt;/A;SPMgt;
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;SPMlt;CAPTION class=;SPMquot;BOTTOM;SPMquot;;SPMgt;;SPMlt;STRONG;SPMgt;Figure:;SPMlt;/STRONG;SPMgt;
Plot for problem 2.;SPMlt;/CAPTION;SPMgt;
;SPMlt;TR;SPMgt;;SPMlt;TD;SPMgt;;SPMlt;IMG
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<DIV class="CENTER">
<tex2html_image_mark>#figure16#
</DIV>;SPMquot;;SPMgt;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;/TABLE;SPMgt;
;SPMlt;/DIV;SPMgt;

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;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;Here is the output from

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;SPMlt;P;SPMgt;
Table[<#19#>x, (Sqrt[x + 25] - 5)/x<#19#>, <#20#>x, -1/5, 1/5, 1/20<#20#>] //N //TableForm
;SPMlt;PRE;SPMgt;
    -0.2    0.100201
    -0.15   0.10015
    -0.1    0.1001
    -0.05   0.10005
    0       Indeterminate
    0.05    0.09995
    0.1     0.0999002
    0.15    0.0998504
    0.2     0.0998008
;SPMlt;/PRE;SPMgt;
It may seem more natural to ``force''  to use decimal arithmetic during the calculation instead of evoking N afterwards, perhaps like this:

<P>
;SPMlt;P;SPMgt;
Table[<#21#>x, (Sqrt[x + 25] - 5)/x<#21#>, <#22#>x, -.2, .2, .05<#22#>] //TableForm.

<P>
;SPMlt;P;SPMgt;
However, with in-exact arithmetic, the user has to be alert enough to reject output that is too close to the singular point:
;SPMlt;PRE;SPMgt;
-0.2             0.100201
-0.15            0.10015
-0.1             0.1001
-0.05            0.10005
           -17
-1.38778 10      0.
0.05             0.09995
0.1              0.0999002
0.15             0.0998504
0.2              0.0998008
;SPMlt;/PRE;SPMgt;

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;If Example 12 seems obscure, ask your instructor for help.
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;As indicated, ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math6#<tex2html_image_mark>#tex2html_wrap_inline289#
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 5 + <I>x</I>/10
 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 179.22ex; vertical-align: -0.52ex; ;SPMquot; SRC=;SPMquot;img5.png;SPMquot;
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;SPMquot;;SPMgt; ;SPMlt;IMG
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;SPMquot;;SPMgt; 5 + ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;/10, so  we can replace the given limit by ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math8#<tex2html_image_mark>#tex2html_wrap_inline296#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 179.22ex; vertical-align: -0.68ex; ;SPMquot; SRC=;SPMquot;img2.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math9#<tex2html_image_mark>#tex2html_wrap_inline298#
;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math10#<tex2html_image_mark>#tex2html_wrap_inline300#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 1.02ex; vertical-align: 161.92ex; ;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math11#<tex2html_image_mark>#tex2html_wrap_inline302#
;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; = 1/10.
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;Running the indicated code gives:
;SPMlt;PRE;SPMgt;
0.1       0.0999002
0.01      0.09999
0.001     0.099999
0.0001    0.0999999
0.00001   0.1
;SPMlt;/PRE;SPMgt;
With x -;SPMamp;gt; a - 1/10;SPMamp;#710;k, we get:
;SPMlt;PRE;SPMgt;
-0.1       0.1001
-0.01      0.10001
-0.001     0.100001
-0.0001    0.1
-0.00001   0.1
;SPMlt;/PRE;SPMgt;

<P>
;SPMlt;P;SPMgt;
Note: We have supplied a function called LimitTable that uses this code as its core.

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;Your opinion is more important than mine.

<P>
;SPMlt;/LI;SPMgt;
;SPMlt;/OL;SPMgt;

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;(2.2.8) ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math12#<tex2html_image_mark>#tex2html_wrap_inline304#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 3.26ex; vertical-align: 159.85ex; ;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math13#<tex2html_image_mark>#tex2html_wrap_inline306#
;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
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 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 2.30ex; vertical-align: 160.42ex; ;SPMquot; SRC=;SPMquot;img9.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math15#<tex2html_image_mark>#tex2html_wrap_inline310#
;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; = 3/4.

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;(2.2.22) ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math16#<tex2html_image_mark>#tex2html_wrap_inline312#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 3.32ex; vertical-align: 159.85ex; ;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math17#<tex2html_image_mark>#tex2html_wrap_inline314#
;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math18#<tex2html_image_mark>#tex2html_wrap_inline316#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 2.30ex; vertical-align: 160.47ex; ;SPMquot; SRC=;SPMquot;img11.png;SPMquot;
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;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; = 1/6.

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;(2.2.32) ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math20#<tex2html_image_mark>#tex2html_wrap_inline320#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 3.32ex; vertical-align: 159.79ex; ;SPMquot; SRC=;SPMquot;img12.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math21#<tex2html_image_mark>#tex2html_wrap_inline322#
;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math22#<tex2html_image_mark>#tex2html_wrap_inline324#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 2.30ex; vertical-align: 160.47ex; ;SPMquot; SRC=;SPMquot;img13.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math23#<tex2html_image_mark>#tex2html_wrap_inline326#
;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; = 4.

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;(2.2.35) ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math24#<tex2html_image_mark>#tex2html_wrap_inline328#

 --;SPMgt;
;SPMlt;IMG
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;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math26#<tex2html_image_mark>#tex2html_wrap_inline332#
<tex2html_image_mark>#tex2html_wrap_inline333#
<tex2html_image_mark>#tex2html_wrap_inline334#
 - <tex2html_image_mark>#tex2html_wrap_inline335#
<tex2html_image_mark>#tex2html_wrap_inline336#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 3.26ex; vertical-align: 159.69ex; ;SPMquot; SRC=;SPMquot;img14.png;SPMquot;
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;SPMquot;;SPMgt;;SPMlt;IMG
 STYLE=;SPMquot;height: 2.94ex; vertical-align: 159.48ex; ;SPMquot; SRC=;SPMquot;img15.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math28#<tex2html_image_mark>#tex2html_wrap_inline340#
<tex2html_image_mark>#tex2html_wrap_inline341#
<tex2html_image_mark>#tex2html_wrap_inline342#
-<tex2html_image_mark>#tex2html_wrap_inline343#
;SPMquot;;SPMgt;;SPMlt;IMG
 STYLE=;SPMquot;height: 3.90ex; vertical-align: 158.87ex; ;SPMquot; SRC=;SPMquot;img16.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math29#<tex2html_image_mark>#tex2html_wrap_inline345#
;SPMquot;;SPMgt; - ;SPMlt;IMG
 STYLE=;SPMquot;height: 3.07ex; vertical-align: 159.88ex; ;SPMquot; SRC=;SPMquot;img17.png;SPMquot;
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;SPMquot;;SPMgt;;SPMlt;IMG
 STYLE=;SPMquot;height: 2.94ex; vertical-align: 159.48ex; ;SPMquot; SRC=;SPMquot;img18.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math31#<tex2html_image_mark>#tex2html_wrap_inline349#
<tex2html_image_mark>#tex2html_wrap_inline350#
<tex2html_image_mark>#tex2html_wrap_inline351#
-<tex2html_image_mark>#tex2html_wrap_inline352#
<tex2html_image_mark>#tex2html_wrap_inline353#
;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; = - 1/10.

<P>
;SPMlt;P;SPMgt;
;SPMlt;/LI;SPMgt;
;SPMlt;LI;SPMgt;(2.2.36) ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math32#<tex2html_image_mark>#tex2html_wrap_inline355#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 3.90ex; vertical-align: 159.27ex; ;SPMquot; SRC=;SPMquot;img19.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math33#<tex2html_image_mark>#tex2html_wrap_inline357#
;SPMquot;;SPMgt; ;SPMlt;FONT SIZE=;SPMquot;+2;SPMquot;;SPMgt;;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math34#<tex2html_image_mark>#tex2html_wrap_inline359#

 --;SPMgt;
;SPMlt;IMG
 STYLE=;SPMquot;height: 2.30ex; vertical-align: 160.47ex; ;SPMquot; SRC=;SPMquot;img20.png;SPMquot;
 ALT=;SPMquot;<tex2html_verbatim_mark>#math35#<tex2html_image_mark>#tex2html_wrap_inline361#
;SPMquot;;SPMgt;;SPMlt;/FONT;SPMgt; = 1/3.
         To do this algebraically is a bit trickey.  Factor the numerator:  ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math36#(<I>x</I><SUP>1/3</SUP> -2)(<I>x</I><SUP>1/3</SUP> + 2)
 --;SPMgt;
(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;1/3;SPMlt;/SUP;SPMgt; -2)(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;1/3;SPMlt;/SUP;SPMgt; + 2)
	 and the denominator: ;SPMlt;!-- MATH
 <tex2html_verbatim_mark>#math37#(<I>x</I><SUP>1/3</SUP> -2)(<I>x</I><SUP>2/3</SUP> +2<I>x</I><SUP>1/3</SUP> + 4)
 --;SPMgt;
(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;1/3;SPMlt;/SUP;SPMgt; -2)(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2/3;SPMlt;/SUP;SPMgt; +2;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;1/3;SPMlt;/SUP;SPMgt; + 4).  Then cancel the common 
	 term and substitute ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt; = 8.
;SPMlt;/LI;SPMgt;
;SPMlt;/OL;SPMgt;
;SPMlt;HR;SPMgt;

<P>
;SPMlt;/BODY;SPMgt;
;SPMlt;/HTML;SPMgt;