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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "Module  8 : Differential Calculus
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 18 "801 :
 Lotsa Limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 17 "O B J E C T I V E" }}{PARA 0 "" 0 "" {TEXT -1 157 "In thi
s project we will examine limits from a numerical, graphical, and symb
olic point of view to better understand the concept and the applicatio
n of limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 9 "S E T U P" }}{PARA 0 "" 0 "" {TEXT -1 253 "In this project
 we will use the following command packages. Type and execute this lin
e before beginning the project below. If you re-enter the worksheet fo
r this project, be sure to re-execute this statement before jumping to
 any point in the worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
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{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
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"_____________________________________________________________________
______________" }}{PARA 4 "" 0 "" {TEXT -1 30 "A. Review of Functions \+
& Plots" }}{PARA 0 "" 0 "" {TEXT -1 83 "______________________________
_____________________________________________________" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 171 "The format for defining a function is a little cumbersom
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{TEXT -1 119 "   -  name of the function\n   -  :=\n   -  independent \+
variable(s)\n   -  crude arrow using a dash and greater than (->)\n" }
{TEXT -1 70 "   -  formula for the function\n   -  semi-colon to termi
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{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________
____________________________________________________________________" 
}}{PARA 4 "" 0 "" {TEXT -1 33 "B. A Geometric Approach To Limits" }}
{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________________
__________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "Now that we've review some bas
ics, lets look at limits from a geometric point of view. The concept o
f limits can seem difficult, but the idea is quite simple when you see
 what is happening geometrically. We will construct a diagram which sh
ows a function and points approaching from the left and right." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "function
 definition                                          a limit target po
int a      the left and right endpoints of an interval near a" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f := x -> 3 + (x-2)*cos((x-2
));   a := 2:   left := -1:   right :=5:" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"$\"\"\"*&,&9$F.\"
\"#!\"\"F.-%$cosG6#F0F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 432 " display( plot( f(x),  x = left..right, color = green),\n    \+
      plot( \{[[a,0],[a,f(a)]],[[0,f(a)],[a,f(a)]] \},  x = left..righ
t,\n            linestyle=3,color = gold, thickness = 2),\n          p
lot([[ a - 1/n, f(a - 1/n)] $n=1..20], x = left..right,\n            s
tyle=point, symbol=circle, color = red),  \n          plot(  [[ a+1/n,
 f(a + 1/n)] $n=1..20], x = left..right,  \n            style=point, s
ymbol=circle, color = blue));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 
300 {PLOTDATA 2 "6)-%'CURVESG6$7S7$$!\"\"\"\"!$\"3[O8!)*[x*pf!#<7$$!3/
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0$F-7$$\"3#)************\\?F-$\"3O[(>I^P*\\IF--F\\[l6&F^[lF_[lF_[lF`[l
FgclF[dl-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAULTG-%%VIEWG6$;F(FfzF][m" 1 
2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 607 "The green curve is the function. The yel
low lines indicate where (a,f(a)) is. The blue dots indicate points on
 f(x) as x approaches a from the right, and the green dots indicate po
ints on f(x) where x is converging to a from the left side. By looking
 at this diagram, you can guess the right limit by looking at what y v
alue the points seem to be converging to. In a similar way, you can gu
ess the left limit by looking at what value the red dots seem to be co
nverging to. If the red and blue dots appear to be converging to the s
ame value, then the limit exists and equals the value they are converg
ing to." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 172 "Here is another example where the f
unction is left and right limits are not the same. After re-defining f
(x), copy and paste the display command block above and re-execute." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 75 "f := x -> Heaviside(x-1) - Heaviside(1-x); a := 1:  left := -2: \+
 right :=4:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)op
eratorG%&arrowGF(,&-%*HeavisideG6#,&9$\"\"\"F2!\"\"F2-F.6#,&F2F2F1F3F3
F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 431 "display( plot( f
(x),  x = left..right, color = green),\n          plot( \{[[a,0],[a,f(
a)]],[[0,f(a)],[a,f(a)]] \},  x = left..right,\n            linestyle=
3,color = gold, thickness = 2),\n          plot([[ a - 1/n, f(a - 1/n)
] $n=1..20], x = left..right,\n            style=point, symbol=circle,
 color = red),  \n          plot(  [[ a+1/n, f(a + 1/n)] $n=1..20], x \+
= left..right,  \n            style=point, symbol=circle, color = blue
));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6)-%'CURVES
G6$7hn7$$!\"#\"\"!$!\"\"F*7$$!3!******\\2<#p=!#<F+7$$!31++D^NUb<F0F+7$
$!36++]K3XF;F0F+7$$!3%)****\\F)H')\\\"F0F+7$$!3#****\\i3@/P\"F0F+7$$!3
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Fhp7$$\"37++]Zn%)o8F0Fhp7$$\"3y******4FL(\\\"F0Fhp7$$\"3#)****\\d6.B;F
0Fhp7$$\"3(****\\(o3lW<F0Fhp7$$\"3!*****\\A))oz=F0Fhp7$$\"3e******Hk-,
?F0Fhp7$$\"36+++D-eI@F0Fhp7$$\"3u***\\(=_(zC#F0Fhp7$$\"3M+++b*=jP#F0Fh
p7$$\"3g***\\(3/3(\\#F0Fhp7$$\"33++vB4JBEF0Fhp7$$\"3u*****\\KCnu#F0Fhp
7$$\"3s***\\(=n#f(GF0Fhp7$$\"3P+++!)RO+IF0Fhp7$$\"30++]_!>w7$F0Fhp7$$
\"3O++v)Q?QD$F0Fhp7$$\"3G+++5jypLF0Fhp7$$\"3<++]Ujp-NF0Fhp7$$\"3++++gE
d@OF0Fhp7$$\"39++v3'>$[PF0Fhp7$$\"37++D6EjpQF0Fhp7$$\"\"%F*Fhp-%'COLOU
RG6&%$RGBG$F*F*$\"*++++\"!\")Faw-F$6&7$7$Faw%%FAILG7$FhpFiw-F^w6&F`w$
\")+++!)Fdw$\")AR!)\\Fdw$\")Vyg>Fdw-%*THICKNESSG6#\"\"#-%*LINESTYLEG6#
\"\"$-F$6&7$7$FhpFawFjwF[xFcxFgx-F$6&767$FawF+7$$\"3++++++++]FIF+7$$\"
3ImmmmmmmmFIF+7$$\"3++++++++vFIF+7$$\"3U+++++++!)FIF+7$$\"3qLLLLLLL$)F
IF+7$$\"3%4dG9dG9d)FIF+7$$\"3+++++++]()FIF+7$$\"3S))))))))))))))))FIF+
7$$\"3A+++++++!*FIF+7$$\"3g!4444444*FIF+7$$\"3Immmmmmm\"*FIF+7$$\"3GJ#
p2Bp2B*FIF+7$$\"3-'G9dG9dG*FIF+7$$\"3[LLLLLLL$*FIF+7$$\"3+++++++v$*FIF
+7$$\"3\"GN#)eqk<T*FIF+7$$\"3?WWWWWWW%*FIF+7$$\"3E:j_5Uot%*FIF+7$$\"3a
*************\\*FIF+-F^w6&F`wFbwFawFaw-%&STYLEG6#%&POINTG-%'SYMBOLG6#%
'CIRCLEG-F$6&767$$FfxF*Fhp7$$\"3++++++++:F0Fhp7$$\"3ELLLLLLL8F0Fhp7$$
\"3+++++++]7F0Fhp7$$\"3%**************>\"F0Fhp7$$\"3ummmmmmm6F0Fhp7$$
\"3zUr&G9dG9\"F0Fhp7$$\"3+++++++D6F0Fhp7$$\"3;66666666F0Fhp7$$\"33++++
+++6F0Fhp7$$\"3#34444444\"F0Fhp7$$\"3ELLLLLL$3\"F0Fhp7$$\"3)o2Bp2Bp2\"
F0Fhp7$$\"3Sr&G9dG92\"F0Fhp7$$\"3mmmmmmmm5F0Fhp7$$\"3++++++]i5F0Fhp7$$
\"3sk<THN#)e5F0Fhp7$$\"3ebbbbbbb5F0Fhp7$$\"3Oot%*y:j_5F0Fhp7$$\"3/++++
++]5F0Fhp-F^w6&F`wFawFawFbwF^]lFb]l-%+AXESLABELSG6%Q\"x6\"Q!6\"%(DEFAU
LTG-%%VIEWG6$;F(F[wF]bl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 276 "In this \+
case, the function is not continuous. It has a jump at x = 1. The red \+
points coming from the left approach -1 while the blue points approach
ing from the right approach +1. In this case we say that the limit is \+
undefined since there is not a single value for the limit." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_______________________
____________________________________________________________" }}{PARA 
4 "" 0 "" {TEXT -1 33 "C. A Numerical Approach To Limits" }}{PARA 0 "
" 0 "" {TEXT -1 83 "__________________________________________________
_________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 343 "Now that
 we've seen a picture of what is happening, lets take those same funct
ions from above and examine them from a numeric point of view. We are \+
going to take values of x that approach a from both the left and right
. The x values approaching from the left will be slightly less than a,
 and the values from the right will be slightly larger." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x \+
-> 3 + (x-2)*cos((x-2));  a :=2:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"$\"\"\"*&,&9$F.\"\"#!\"\"F
.-%$cosG6#F0F.F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "a
rray( [seq([ evalf( a - 10^(-k), 15), evalf(f( a - 10^(-k)), 15)],k = \+
1..12)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7.7$$\"0+++++
+!>!#9$\"0?sMe*\\+HF*7$$\"0++++++*>F*$\"0$e***\\++*HF*7$$\"0+++++!**>F
*$\"0++0++!**HF*7$$\"0+++++***>F*$\"0]++++***HF*7$$\"0++++!****>F*$\"0
++++!****HF*7$$\"0++++*****>F*$\"0++++*****HF*7$$\"0+++!******>F*$\"0+
++!******HF*7$$\"0+++*******>F*$\"0+++*******HF*7$$\"0++!********>F*$
\"0++!********HF*7$$\"0++*********>F*$\"0++*********HF*7$$\"0+!*******
***>F*$\"0+!**********HF*7$$\"0+***********>F*$\"0+***********HF*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "array( [seq([ evalf( a + 10^
(-k), 15), evalf(f( a + 10^(-k)), 15)],k = 1..12)]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%'matrixG6#7.7$$\"0++++++5#!#9$\"0!y_;/]*4$F*7$$\"0
++++++,#F*$\"0</+]***4IF*7$$\"0+++++5+#F*$\"0++&*****4+$F*7$$\"0+++++,
+#F*$\"0]*******4+IF*7$$\"0++++5++#F*$\"0++++5++$F*7$$\"0++++,++#F*$\"
0++++,++$F*7$$\"0+++5+++#F*$\"0+++5+++$F*7$$\"0+++,+++#F*$\"0+++,+++$F
*7$$\"0++5++++#F*$\"0++5++++$F*7$$\"0++,++++#F*$\"0++,++++$F*7$$\"0+5+
++++#F*$\"0+5+++++$F*7$$\"0+,+++++#F*$\"0+,+++++$F*" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 455 "Here is the same funct
ion we saw above when we graphed it.  This first array creates a list \+
of x values and y values which show what is happening as we approach f
rom the left.. You can see that the x values are less than 2 but getti
ng closer and closer to 2. The second array creates a list of x values
 and y values which show what happens as we approach from the right. Y
ou can see that the x values are greater than 2 but getting closer and
 closer to 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 325 "The y values are approac
hing the value 3 from both the left and the right. From this evidence,
 we can guess that the limit is 3. Lets look at another case. (All of \+
the functions in this section were defined in the previous section and
 you can save some time by copying and pasting them from above rather \+
than re-typing them.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> piecewise(x < 3, -5+x, x <= 6, \+
20-x^2, x < 10, 2*x +1.1);  a:= 6:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6(29$\"\"$,&!\"&
\"\"\"F0F41F0\"\"',&\"#?F4*$)F0\"\"#F4!\"\"2F0\"#5,&F0F;$\"#6F<F4F(F(F
(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 368 "Co
py and paste the two array commands above, and re-execute them to crea
te left and right approaches to the limit for this function. In this c
ase the left and right limits are different. From the left, the value \+
is 16, while from the right, the value is 13. When the left and right \+
limits disagree, we say that the limit does not exist, since there is \+
no single limit." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "_______________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 32 "D. A Symbolic App
roach To Limits" }}{PARA 0 "" 0 "" {TEXT -1 83 "______________________
_____________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 192 "We've approached limits from a geometric and a
 numeric point of view. Lets now address the symbolic side. Maple is a
ble to compute the left limit, the right limit, and the limit of a fun
ction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "restart:\nf := x -> 3 + (x-2)*cos((x - 2));  a := 2:
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&ar
rowGF(,&\"\"$\"\"\"*&,&9$F.\"\"#!\"\"F.-%$cosG6#F0F.F.F(F(F(" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 149 "- The Li
mit command with a capital \"L\" displays the limit in stardard mathem
atical notation. To find a limit just type the function and target val
ue." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "-
 The \"%\" character is a reference to the most recent result processe
d by Maple, in this case the Limit command." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "-The " }{TEXT 256 5 "value" }
{TEXT -1 41 " command computes the value of the limit." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "- For left or right \+
limits include the option third parameter \"left\" or \"right\"." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "Limit( f(x), x = a, left): % = value(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%&LimitG6%,&\"\"$\"\"\"*&,&%\"xGF)\"\"#!\"\"F)-%$cosG
6#F+F)F)/F,F-%%leftGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "L
imit( f(x), x = a, right): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%&LimitG6%,&\"\"$\"\"\"*&,&%\"xGF)\"\"#!\"\"F)-%$cosG6#F+F)F)/
F,F-%&rightGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Limit( f(
x), x = a): % = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&Limit
G6$,&\"\"$\"\"\"*&,&%\"xGF)\"\"#!\"\"F)-%$cosG6#F+F)F)/F,F-F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "3 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }