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{SECT 0 {PARA 258 "" 0 "" {TEXT -1 6 "LIMITS" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 
18 "Calculus I Project" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 2 "  " }{TEXT 
258 11 "Objectives:" }}{PARA 0 "" 0 "" {TEXT -1 71 " To investigate th
e limit of a function at a given point numerically.  " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Using a computer alge
bra system you will be able to create a table of values of x and f(x) \+
by constructing a 'loop'." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 15 "Solved Example:" }}
{PARA 0 "" 0 "" {TEXT -1 33 "Investigate the limit of  f(x) = " }
{XPPEDIT 18 0 "(sqrt(x+25)-5)/x;" "6#*&,&-%%sqrtG6#,&%\"xG\"\"\"\"#DF*
F*\"\"&!\"\"F*F)F-" }{TEXT -1 13 "   as x -> 0." }}}{PARA 0 "" 0 "" 
{TEXT -1 3 "   " }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 260 
0 "" }{TEXT 261 11 "Solution:  " }}{PARA 3 "" 0 "" {TEXT 262 38 "This \+
problem is done in several steps." }}{SECT 0 {PARA 0 "" 0 "" {TEXT -1 
55 "  We first discuss the right limit of f(x) as x -> 0.  " }}{PARA 
0 "" 0 "" {TEXT -1 4 "    " }}{SECT 0 {PARA 0 "" 0 "" {TEXT 271 0 "" }
{TEXT 256 21 " Initialize 'a' and '" }{TEXT 270 3 "h'." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a:=0;  \+
h:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"hG\"\"\"" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 
1 " " }{TEXT 256 28 "Define x as a function of n." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x:=n-> a+h/5
^n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGf*6#%\"nG6\"6$%)operatorG
%&arrowGF(,&%\"aG\"\"\"*&%\"hGF.)\"\"&9$!\"\"F.F(F(F(" }}}}{SECT 0 
{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 265 0 "" }{TEXT 266 0 "" }{TEXT 
256 28 "Define f as a function of x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x->((x+25)^.5-5)/x;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(*&,&*$),&9$\"\"\"\"#DF2$\"\"&!\"\"F2F2F5F6F2F1F6F(F(F(" }}}}{SECT 
0 {PARA 0 "" 0 "" {TEXT -1 10 " We evalua" }{TEXT 268 0 "" }{TEXT -1 
14 "te f(x) at x =" }{TEXT 257 0 "" }{TEXT 256 6 "  0 + " }{XPPEDIT 
18 0 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT -1 8 " ,  0 + " }{XPPEDIT 
18 0 "1/(5^2);" "6#*&\"\"\"F$*$\"\"&\"\"#!\"\"" }{TEXT -1 7 ",  0 + " 
}{XPPEDIT 18 0 "1/(5^3);" "6#*&\"\"\"F$*$\"\"&\"\"$!\"\"" }{TEXT -1 7 
" , ... " }{TEXT 267 2 "  " }}{PARA 0 "" 0 "" {TEXT 269 1 " " }{TEXT 
275 0 "" }{TEXT -1 0 "" }{TEXT 256 69 "by creating a table for differe
nt values of (x, f(x))  using  a loop;" }{TEXT 274 2 "  " }}{PARA 0 "
" 0 "" {TEXT 276 1 " " }{TEXT 273 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "for n from 1 by 1 to 8 do\nprint( evalf(x(n)),`      \+
          `,evalf(f(x(n))) );\nod;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6%$\"+++++?!#5%1~~~~~~~~~~~~~~~~G$\")&z+)**!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%$\"+++++S!#6%1~~~~~~~~~~~~~~~~G$\")D+'***!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%$\"+++++!)!#7%1~~~~~~~~~~~~~~~~G$\")+?****!
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+++++;!#7%1~~~~~~~~~~~~~~~~G
$\")D\")****!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+++++K!#8%1~~~~
~~~~~~~~~~~~G$\"*++++\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"++++
+k!#9%1~~~~~~~~~~~~~~~~G$\"*++++\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%$\"++++!G\"!#9%1~~~~~~~~~~~~~~~~G$\"*++++\"!\"*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6%$\"++++gD!#:%1~~~~~~~~~~~~~~~~G$\"*++++\"!\"*" }}}
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 "
 " }{TEXT 263 51 "We then discuss the left limit of f(x) as x -> 0.  \+
" }}{PARA 0 "" 0 "" {TEXT 264 26 "We now take numbers for x " }{TEXT 
-1 12 "such as 0 - " }{XPPEDIT 18 0 "1/5;" "6#*&\"\"\"F$\"\"&!\"\"" }
{TEXT -1 8 ",   0 - " }{XPPEDIT 18 0 "1/(5^2);" "6#*&\"\"\"F$*$\"\"&\"
\"#!\"\"" }{TEXT -1 58 ", etc. to the left of 0 and evaluate f(x) at t
hese values." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a:= 0; h:= -
1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hG!\"\"" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG!\"\"" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "x:=n-> a+h/5^n;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"xGf*6#%\"nG6\"6$%)operatorG%&arrowGF(,&%\"aG\"\"\"*
&%\"hGF.)\"\"&9$!\"\"F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "f:=x->((x+25)^.5-5)/x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
f*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&*$),&9$\"\"\"\"#DF2$\"\"&!\"\"F
2F2F5F6F2F1F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " for
 n from 1 by 1 to 8 do\nprint( evalf(x(n)),`                `,evalf(f(
x(n))) );\nod;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+++++?!#5%1~~~~~
~~~~~~~~~~~G$\"*03?+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+++++S
!#6%1~~~~~~~~~~~~~~~~G$\"*D+/+\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6%$!+++++!)!#7%1~~~~~~~~~~~~~~~~G$\"*+!3+5!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%$!+++++;!#7%1~~~~~~~~~~~~~~~~G$\"*v=++\"!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%$!+++++K!#8%1~~~~~~~~~~~~~~~~G$\"*++++\"!\"*
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+++++k!#9%1~~~~~~~~~~~~~~~~G$\"*
++++\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!++++!G\"!#9%1~~~~~~~~~
~~~~~~~G$\"*++++\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!++++gD!#:%
1~~~~~~~~~~~~~~~~G$\"*++++\"!\"*" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 1 " " }{TEXT 277 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 30 "The
 two tables suggest  that  " }{XPPEDIT 18 0 "limit((sqrt(x+25)-5)/x,x \+
= 0);" "6#-%&limitG6$*&,&-%%sqrtG6#,&%\"xG\"\"\"\"#DF-F-\"\"&!\"\"F-F,
F0/F,\"\"!" }{TEXT -1 9 " =   .1  " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 75 "_______________________________________
____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 259 "" 0 "" {TEXT -1 13 "  ASSIGNMENT " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 278 8 "P
roblem:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
55 "Using above methods , investigate the limit of  f(x) = " }
{XPPEDIT 18 0 "(x^1.5-8)/(x-8);" "6#*&,&)%\"xG-%&FloatG6$\"#:!\"\"\"\"
\"\"\")F+F,,&F&F,F-F+F+" }{TEXT -1 22 ",  as x approaches  4." }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 " ________
__________________________________________________________________   \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 161 "MSIP
 Grant #P120A80089-98:  \"Three Urban Calculus Reform programs: Adopti
ng the Best\" 1998-2001                                               \+
                   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{MARK "10 5 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }