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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 18 "Limits Student Lab" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "
Notes to Student:" }}{PARA 0 "" 0 "" {TEXT -1 129 "This lab has many a
nimations to help you get an intuitive feel for the limiting process. \+
It is important for you to focus on the " }{TEXT 337 7 "dynamic" }
{TEXT -1 270 " nature of the lab. Limits are about movement and how a \+
change in one value affects another value. Calculus has been often des
cribed as the mathematics of change. Keep that in mind throughout this
 lab for it is crucial in laying down a good foundation for later topi
cs. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Introduction:" }}{PARA 
0 "" 0 "" {TEXT -1 183 "The limiting process is at the very heart of c
alculus and without it the foundation of calculus quickly unravels.The
 animations of this lab are all designed to help you visualize the " }
{TEXT 338 16 "limiting process" }{TEXT -1 257 ". This process hinges o
n examining how a change in one quantity affects another quantity. How
 functions behave as we alter permissable domain values will be explor
ed closely throughout the lab and you should focus on the big picture \+
concept of each section." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Obje
ctives Defined:" }}{PARA 258 "" 0 "" {TEXT -1 141 "The purpose of this
 module is to improve the student's conceptual and computational under
standing of limits. These goals will be achieved by " }}{PARA 258 "" 
0 "" {TEXT -1 129 "1. Demonstrating the informal definition of limits \+
through graphical representations and converging/diverging         seq
uences. " }}{PARA 258 "" 0 "" {TEXT -1 50 "2. Exploring when limits fa
il to exist at a point." }}{PARA 258 "" 0 "" {TEXT -1 93 "3. Demonstra
ting through animation the usage and elements of the \"Squeeze/Sandwhi
ch Theorem\"." }}{PARA 258 "" 0 "" {TEXT -1 45 "4. Examining the preci
se definition of limit." }}{PARA 258 "" 0 "" {TEXT -1 34 "5. Providing
 student problem sets." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "The Pr
ocess of Limiting Values (Obj.1): " }{TEXT 264 8 "Animated" }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "                                  \+
                                " }{TEXT 257 0 "" }{TEXT 258 16 "Two S
ided Limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" 
{TEXT -1 53 "When one discusses limits one is really describing a " }
{TEXT 256 0 "" }{TEXT -1 251 "covariational process between inputed va
lues and their respective outputs. If we are interested in how a funct
ion's outputs behave as we approach a designated input value then we a
re describing a limiting process. Consider the following limit problem
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "    \+
                                                                      \+
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{PARA 258 "" 0 "" {TEXT -1 270 "Although the problem is not defined at
 x = 1, we can still determine the limit as discussed in class. The an
imation below will give you a graphical look at the limiting process. \+
Pay particular note to the left and right-hand limit sequences. Place \+
the cursor in the red " }{TEXT 286 7 "restart" }{TEXT -1 146 " code wo
rd and press \"Enter\". Click on the picture and use the animation con
trols at the top to run the animation. BE PATIENT for the code to set.
" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "restart:with(plots):with(plottools):\nDigits:=5:\nf:=
x->(x^2-1)/(x-1):" }{TEXT -1 31 "*/ Function examined (graph 1)." }
{MPLTEXT 1 0 1472 "\nh:=0.9:\nn:=50:\na:=1.0005:\ngraph:=display(point
plot(\{[a,f(a)],[a,0]\},symbol=circle,symbolsize=15),plot(f(x),x=(a-(h
+signum(h)*1))..(a+(h+signum(h)*1)),thickness=3,color=green)):\nrightp
t:=display(seq(pointplot([a+(n-i)/(n/h),0],symbol=circle,symbolsize=18
,color=blue),i=0..n-1),insequence=true):\nleftpt:=display(seq(pointplo
t([a-(n-i)/(n/h),0],symbol=circle,symbolsize=18,color=blue),i=0..n-1),
insequence=true):\nfright:=display(seq(textplot([a+1.9,2,cat(\"f(x)rig
ht = \",convert(f(a+(n-i)/(n/h),4),string))],color=BLUE,font=[TIMES,RO
MAN,12]),i=0..n-1),insequence=true):\nfleft:=display(seq(textplot([a-1
.9,2,cat(\"f(x)left= \",convert(f(a-(n-i)/(n/h),4),string))],color=BLU
E,font=[TIMES,ROMAN,12]),i=0..n-1),insequence=true):\nxleft:=display(s
eq(textplot([0.7,-1,cat(\"x-left = \",convert(evalf(a-(n-i)/(n/h),4),s
tring))],color=BLUE,font=[TIMES,ROMAN,12]),i=0..n-1),insequence=true):
\nxright:=display(seq(textplot([2,-1,cat(\"x-right=\",convert(evalf(a+
(n-i)/(n/h),4),string))],color=BLUE,font=[TIMES,ROMAN,12]),i=0..n-1),i
nsequence=true):\ncright:=display(seq(ellipse([a+(n-i)/(n/h),f(a+(n-i)
/(n/h))], 0.08,0.1,color=red,filled=true),i=0..n-1),insequence=true):
\ncleft:=display(seq(ellipse([a-(n-i)/(n/h),f(a-(n-i)/(n/h))],0.08,0.1
,color=red,filled=true),i=0..n-1),insequence=true):\nheading:=display(
textplot([1,3.5,`Two-Sided Limit`],color=magenta,font=[TIMES,ROMAN,14]
)):\ndisplay(cright,cleft,graph,xleft,xright,rightpt,leftpt,fright,fle
ft,heading):Digits:=5:\ng:=x->sqrt(x,symbolic):" }{TEXT -1 31 "*/ Func
tion examined (graph 2)." }{MPLTEXT 1 0 961 " \nh:=0.5:\na:=0.0000005:
\ngraph2:=display(pointplot(\{[a,g(a)],[a,0]\},symbol=circle,symbolsiz
e=15),plot(g(x),x=0..(a+(h+signum(h)*1)),thickness=3,color=green)):\nr
ightpt2:=display(seq(pointplot([a+(n-i)/(n/h),0],symbol=circle,symbols
ize=18,color=blue),i=0..n-1),insequence=true):\nfright2:=display(seq(t
extplot([a+0.8,2,cat(\"g(x)=\",convert(g(a+(n-i)/(n/h),4),string))],co
lor=BLUE,font=[TIMES,ROMAN,12]),i=0..n-1),insequence=true):\nxright2:=
display(seq(textplot([0.8,-1,cat(\"x-right=\",convert(evalf(a+(n-i)/(2
*n/h),4),string))],color=BLUE,font=[TIMES,ROMAN,12]),i=0..n-1),inseque
nce=true):\ncright2:=display(seq(ellipse([a+(n-i)/(n/h),g(a+(n-i)/(n/h
))], 0.03,0.06,color=red,filled=true),i=0..n-1),insequence=true):\nhea
ding2:=display(textplot([0.8,3.2,`One-Sided Limit (Square Root Functio
n)`],color=magenta,font=[TIMES,ROMAN,14])):\ndisplay(cright2,graph2,xr
ight2,rightpt2,fright2,heading2):\ndisplay(cright,cleft,graph,xleft,xr
ight,rightpt,leftpt,fright,fleft,heading);" }{TEXT -1 0 "" }}}{EXCHG 
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 116 "Notice that the function has a limit value of 2 even though f(
1) is not defined because x = 1 is not in the domain. " }}{PARA 0 "" 
0 "" {TEXT -1 35 "The above example demonstrates the " }{TEXT 355 28 "
informal definition of limit" }{TEXT -1 34 " and can be expressed in w
ords as:" }}{PARA 0 "" 0 "" {TEXT 348 0 "" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 7 "       " }{TEXT 349 1 "\"" }{TEXT 345 31 " The limi
t of f(x) at a point ," }{TEXT 351 2 " a" }{TEXT 344 46 ", is a finite
 number, L, if as x gets close to" }{TEXT 350 0 "" }{TEXT 346 2 " a" }
{TEXT 347 24 ", f(x) gets close to L\"." }}{PARA 258 "" 0 "" {TEXT -1 
0 "" }}{PARA 258 "" 0 "" {TEXT -1 30 "We write this symbolically as:" 
}}{PARA 257 "" 0 "" {OLE 1 4612 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>
N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::
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::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ:
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:::::::5:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 352 300 "If as we let x approach a value \+
\"a\" the left-hand and right-hand function outputs approach the same \+
value, then we say the limit is that common value. As mentioned in the
 above example \"a\" need not be in the domain of the function. We are
 concerned about how the heights of the function behave for x " }
{TEXT 353 4 "near" }{TEXT 354 287 " \"a\". Limits fail to exist when t
he left-hand function heights do not approach the same value as the ri
ght-hand function heights. They also fail to exist if the function inc
reases/decreases without bound (i.e. gets infinitely large or small) o
r the function oscillates like in sin(1/x). " }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "
" {TEXT 260 16 "One-Sided Limits" }{TEXT 282 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 118 "Many functions have a r
estricted domain. This can create what we call \"one-sided limits\". P
lace the cursor in the red " }{TEXT 287 7 "display" }{TEXT 288 97 " an
d press \"Enter\". Once agian, you can use the animation toolbox after
 you click on the picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 58 "display(cright2,graph2,xright2,rightpt2,fri
ght2,heading2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "When Limits Fail to Exist (Obj.2)
: " }{TEXT 261 8 "Animated" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
257 "" 0 "" {TEXT -1 0 "" }{TEXT 283 0 "" }{TEXT 284 0 "" }{TEXT 285 
25 "When Limits Fail to Exist" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 90 "The following are several examples of whe
n limits fail to exist. Examine each one closely." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 297 0 "" }{TEXT 300 0 "" }
{TEXT 301 0 "" }{TEXT 302 35 "Jump Discontinuity Function Example" }
{TEXT 298 0 "" }{TEXT 299 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 295 144 "Some limits fail to exist because the f
unction has a jump discontinuity near the point of question. Consider \+
the piecewise function defined as  " }}{PARA 0 "" 0 "" {TEXT -1 60 "  \+
                                                          " }{OLE 1 
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<<:Z=\\>RSHe:rFDZ;<::::::>Z:b:TC>:Q:G;Sjysy=J`B:;JB<Z:VY[B:IZ:>:sg:<J:
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:::::::::::::::::::::::::::::::1:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 316 28 "Place the cursor in the red " }{TEXT 
296 7 "restart" }{TEXT 303 85 " and press \"Enter\". You can use the a
nimation toolbox after you click on the picture." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1728 "restart:wi
th(plots):with(plottools):Digits:=5:\ng:=x->1/x^2:\nf:=x->piecewise(x<
3,-x+2,x>=3,sin(x)+6):\nh:=1:b:=3:\nn:=50:\na:=0.0005:\ngraph1:=displa
y(pointplot(\{[a+3,f(a+3)],[a+3,0]\},symbol=circle,symbolsize=15),plot
(f(x),x=b+a-(b+signum(b)*1)..(b+a+(b+signum(b)*1)),thickness=3,color=b
lue,view=[0..10,-1..10],title=\"Limit Fails to Exist\\nFunction has Ju
mp Discontinuity\",discont=true,titlefont=[TIMES,BOLD,16])):\nrightpt1
:=display(seq(pointplot([3+a+(n-i)/(n/b),0],symbol=circle,symbolsize=1
8,color=red),i=0..n-1),insequence=true):\nleftpt1:=display(seq(pointpl
ot([3+a-(n-i)/(n/b),0],symbol=circle,symbolsize=18,color=red),i=0..n-1
),insequence=true):\ncright1:=display(seq(ellipse([3+a+(n-i)/(n/b),f(3
+a+(n-i)/(n/b))], 0.08,0.3,color=black,filled=true),i=0..n-1),insequen
ce=true):\ncleft1:=display(seq(ellipse([3+a-(n-i)/(n/b),f(3+a-(n-i)/(n
/b))], 0.08,0.3,color=black,filled=true),i=0..n-1),insequence=true):\n
graph2:=display(pointplot(\{[a,g(a)],[a,0]\},symbol=circle,symbolsize=
15),plot(g(x),x=a-(h+signum(h)*1)..(a+(h+signum(h)*1)),thickness=3,col
or=green,view=[-1..1,-1..100],title=\"Limit Fails to Exist\\nF(x) Unbo
unded\",titlefont=[TIMES,BOLD,16])):\nrightpt2:=display(seq(pointplot(
[a+(n-i)/(n/h),0],symbol=circle,symbolsize=18,color=blue),i=0..n-1),in
sequence=true):\nleftpt2:=display(seq(pointplot([a-(n-i)/(n/h),0],symb
ol=circle,symbolsize=18,color=blue),i=0..n-1),insequence=true):\ncrigh
t2:=display(seq(ellipse([a+(n-i)/(n/h),g(a+(n-i)/(n/h))], 0.04,1.2,col
or=black,filled=true),i=0..n-1),insequence=true):\ncleft2:=display(seq
(ellipse([a-(n-i)/(n/h),g(a-(n-i)/(n/h))], 0.04,1.2,color=black,filled
=true),i=0..n-1),insequence=true):\ndisplay(cright2,cleft2,graph2,left
pt2,rightpt2):\ndisplay(cright1,cleft1,graph1,leftpt1,rightpt1);\n" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "The next example shows ....\n" }}{PARA 257 "" 0 "" {TEXT 
304 0 "" }{TEXT 307 0 "" }{TEXT 308 0 "" }{TEXT 309 26 "Unbounded Func
tion Example" }{TEXT 305 0 "" }{TEXT 306 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 101 "Some limits fail to exist bec
ause the function is unbounded near the point of question. The animati
on" }{TEXT -1 1 " " }{TEXT 312 75 "below shows a function where the li
mit does not exist because the function," }{TEXT -1 1 " " }{TEXT 314 
7 "f(x) = " }{XPPEDIT 315 0 "1/(x^2);" "6#*&\"\"\"F$*$%\"xG\"\"#!\"\"
" }{TEXT -1 1 " " }{TEXT 313 87 " grows without bound as x approaches \+
zero from both sides. Place the cursor in the red " }{TEXT 293 7 "disp
lay" }{TEXT 294 97 " and press \"Enter\". Once agian, you can use the \+
animation toolbox after you click on the picture." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display(cright2,cle
ft2,graph2,leftpt2,rightpt2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 310 19 "Student Problem Set" 
}{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 262 10 "Directions" }{TEXT 
311 1 ":" }{TEXT 263 96 " Write your answers in the space available. Y
ou may use a calculator to aid in your exploration." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "#1. What does" }{TEXT 
327 9 " f(a) = L" }{TEXT -1 6 " mean?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "#2. What \+
does" }{XPPEDIT 18 0 "limit(f(x) = L,x = a);" "6#-%&limitG6$/-%\"fG6#%
\"xG%\"LG/F*%\"aG" }{TEXT -1 7 "  mean?" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "#3. How d
o you informally decide if a limit at a point exists? " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 36 "#4. Create a function that satisfies" }{XPPEDIT 18 0 "limit(f(x
),x = a);" "6#-%&limitG6$-%\"fG6#%\"xG/F)%\"aG" }{TEXT -1 60 "   =   L
,  but the function is not defined for the point x =" }{TEXT 266 2 " a
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
109 "#. 5 Determine the following limits by examining a table of value
s approaching from both sides of the point, " }{TEXT 326 1 "a" }{TEXT 
-1 68 ". Be ready to discuss the graph in general and defend your anal
ysis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {OLE 1 2660 
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{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Squeeze Theorem (Obj.3):  " }
{TEXT 259 8 "Animated" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 
0 "" {TEXT 267 19 "The Squeeze Theorem" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 268 154 "The Squeeze/Sandwich Theorem is a t
echnique to determine limits by bounding the given function, f(x), bot
h above and below by some other known functions. " }}{PARA 0 "" 0 "" 
{TEXT 289 27 "The example coded below has" }{TEXT -1 1 " " }{XPPEDIT 
273 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT 274 4 "sin(" }
{XPPEDIT 275 0 "Pi/x;" "6#*&%#PiG\"\"\"%\"xG!\"\"" }{TEXT 276 2 " )" }
{TEXT 291 1 " " }{TEXT -1 1 " " }{TEXT 269 13 "bounded above" }{TEXT 
-1 1 " " }{TEXT 270 2 "by" }{TEXT -1 1 " " }{TEXT 277 8 "p1(x) = " }
{XPPEDIT 278 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 1 " " }{TEXT 271 20 "
and bounded below by" }{TEXT -1 1 " " }{TEXT 279 10 "p2(x) = -(" }
{XPPEDIT 280 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT 281 1 ")" }{TEXT 290 1 "
." }{TEXT -1 1 " " }{TEXT 272 149 "The animation shows that as x  appr
oaches zero, the oscillating f(x) is \"squeezed/sandwiched\" by the bo
unding functions. Place the cursor in the red " }{TEXT 328 7 "restart
" }{TEXT 329 19 " and press \"Enter\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1175 "restart:with(plots):wi
th(plottools):\nDigits:=5:\nf:=x->x^2*sin(Pi/x):p1:=x->x^2:p2:=x->-(x^
2):\nh:=1:\nn:=50:\na:=0.0005:\nb:=a^2*sin(Pi/a):\ngraph:=display(plot
(p1(x),x=-1..1,thickness=2),plot(p2(x),x=-1..1,thickness=2),plot(f(x),
x=(a-(h+signum(h)*1))..(a+(h+signum(h)*1)),thickness=3,color=green,vie
w=[-1...1,-1.5..1.5],title=\"SqueezeTheorem\",titlefont=[TIMES,ROMAN,1
4])):cright:=display(seq(ellipse([a+(n-i)/(n/h),f(a+(n-i)/(n/h))],0.02
,0.04,color=black,filled=true),i=0..n-1),insequence=true):\ncleft:=dis
play(seq(ellipse([a-(n-i)/(n/h),f(a-(n-i)/(n/h))],0.02,0.04,color=blac
k,filled=true),i=0..n-1),insequence=true):\np1left:=display(seq(ellips
e([a-(n-i)/(n/h),p1(a-(n-i)/(n/h))],0.02,0.04,color=black,filled=true)
,i=0..n-1),insequence=true):\np2left:=display(seq(ellipse([a-(n-i)/(n/
h),p2(a-(n-i)/(n/h))],0.02,0.04,color=black,filled=true),i=0..n-1),ins
equence=true):\np1right:=display(seq(ellipse([a+(n-i)/(n/h),p1(a+(n-i)
/(n/h))],0.02,0.04,color=black,filled=true),i=0..n-1),insequence=true)
:\np2right:=display(seq(ellipse([a+(n-i)/(n/h),p2(a+(n-i)/(n/h))],0.02
,0.04,color=black,filled=true),i=0..n-1),insequence=true):\ndisplay(cr
ight,p1left,cleft,graph,p2left,p1right,p2right);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Epsilon-
Delta Animation (Obj.4): " }{TEXT 336 8 "Animated" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The formal definition of \+
limit is expressed in terms of absolute values and is commonly written
 as:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " \+
            " }{TEXT 330 4 "Let " }{TEXT 319 4 "f(x)" }{TEXT 331 66 " \+
be defined for all x in some open interval containing the number " }
{TEXT 317 1 "a" }{TEXT 332 11 ", with the " }}{PARA 258 "" 0 "" {TEXT 
-1 37 "             possible exception that " }{TEXT 320 4 "f(x)" }
{TEXT -1 24 " need not be defined at " }{TEXT 318 3 "a. " }{TEXT -1 
10 " We write:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "                \+
              " }}{PARA 0 "" 0 "" {TEXT -1 46 "                       \+
                       " }{OLE 1 3660 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j
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:::::::::::::::::::::::::::::::::::::::5:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "         " }{TEXT 333 21 " if given
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18 0 "delta;" "6#%&deltaG" }{TEXT -1 5 " > 0 " }{TEXT 335 9 "such that
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "      " }}{PARA 0 "" 
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:::::::::::::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 
0 "" {TEXT -1 198 "We can animate the limiting process and observe the
 relationships between delta, epsilon, and the curve. The two bands yo
u see help you focus in on the area of interest. Place the cursor in t
he red " }{TEXT 321 8 "restart " }{TEXT -1 69 "command and press \"Ent
er\" to obtain the figure upon which to animate." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1346 "restart:w
ith(plots):with(plottools):\nf:=1/x;\na:=2:\nL:=limit(f,x=a):\nW:=1.8:
\nepsilon:=0.5:\nstep:=.8:\nN:=40:\nepsilonmin:=step^N*epsilon:\nleft:
=a-W:\nright:=a+W:\ntop:=evalf(L+W):\nbottom:=evalf(L-W):spread:=50:k:
='k':\nfor k from 1 to N do\n     s:= solve(abs(f-L)=epsilon):Num_root
s:=nops([s]):\n     for i from 1 to Num_roots do\n         if type(s[i
],float) then\n            spread:=spread,abs(s[i]-a):\n             f
i:od:\ndelta:=evalf(min(spread)):\npf:=plot([L+epsilon,f,L-epsilon],x=
left..right,color=[red,black,red],thickness=[1,3,1]):\nfright:=textplo
t([-1.5,0.9,cat(\"L+epsilon = \",convert(evalf(L+epsilon,4),string))],
color=red,font=[TIMES,BOLD,12]): fleft:=textplot([-1.5,.2,cat(\"L-epsi
lon = \",convert(evalf(L-epsilon,4),string))],color=red,font=[TIMES,BO
LD,12]):\ndeltagap:=textplot([4.2,-0.4,cat(\"delta = \",convert(evalf(
min(spread),4),string))],color=magenta,font=[TIMES,BOLD,12]):\np1:=ine
qual(\{y<=L+epsilon,y>=L-epsilon\},x=left..right,y=bottom..top,options
excluded=(color=white),optionsfeasible=(color=blue)):\np2:=inequal(\{x
>=a-delta,x<=a+delta\},x=a-delta..a+delta,y=bottom..top,optionsfeasibl
e=(color=yellow),optionsexcluded = (color=white)):\np[k]:=plots[displa
y](deltagap,fright,fleft,pf,p2,p1,view=[-3...6,-1..2]):\nepsilon:=step
*epsilon:\nod:\nplots[display](p[j]$j=2..N,insequence=true,title=\"Eps
ilon-Delta\",titlefont=[TIMES,BOLD,14]);\n     \n" }}{PARA 0 "" 0 "" 
{TEXT 339 24 "Notice how the change in" }{TEXT -1 1 " " }{XPPEDIT 341 
0 "epsilon;" "6#%(epsilonG" }{TEXT -1 1 " " }{TEXT 340 26 "will result
 in a change in" }{TEXT -1 1 " " }{XPPEDIT 342 0 "delta;" "6#%&deltaG
" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 35 "                                   " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 322 20 "Student Problem S
et." }}{PARA 0 "" 0 "" {TEXT 323 10 "Directions" }{TEXT 325 1 ":" }
{TEXT 324 96 " Write your answers in the space available. You may use \+
a calculator to aid in your exploration." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "#1. Explain with proper grammer an
d mathematical symbolism what  " }{OLE 1 4156 1 "[xm]Br=WfoRrB:::wk;ny
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:::::::::::::::::::1:" }}{PARA 0 "" 0 "" {TEXT -1 12 "      means." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 26 "#2. Consider the function " }{XPPEDIT 18 0 "f(x) = 2*x+3.
;" "6#/-%\"fG6#%\"xG,&*&\"\"#\"\"\"F'F+F+$\"\"$\"\"!F+" }{TEXT -1 15 "
 Find a number " }{XPPEDIT 19 1 "delta;" "6#%&deltaG" }{TEXT -1 11 " s
uch that " }{XPPEDIT 18 0 "abs(f(x)-5);" "6#-%$absG6#,&-%\"fG6#%\"xG\"
\"\"\"\"&!\"\"" }{TEXT -1 16 " < 0.2 whenever " }{XPPEDIT 18 0 "abs(x-
1);" "6#-%$absG6#,&%\"xG\"\"\"F(!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 19 
1 "delta;" "6#%&deltaG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 62 "#3. Prove the following limits using the epsilon-del
ta method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 3 "A. " }{XPPEDIT 18 0 "limit(4*x+5,x = 3);" "6#-%&limitG6$,&*&\"\"
%\"\"\"%\"xGF)F)\"\"&F)/F*\"\"$" }{TEXT -1 80 " = 17                  \+
                                                     B.  " }{XPPEDIT 
18 0 "limit(x^2,x = 2);" "6#-%&limitG6$*$%\"xG\"\"#/F'F(" }{TEXT -1 5 
" = 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "#4. Suppose a manufactur
er  wishes  to create a disk with area 500 square inches within an err
or tolerance of 5 square inches either too large or too small. What is
 an appropriate delta value for the radius?" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 15 "#5. Prove that " }{XPPEDIT 18 0 "limit(1/x,x = 2);" 
"6#-%&limitG6$*&\"\"\"F'%\"xG!\"\"/F(\"\"#" }{TEXT -1 3 " = " }
{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
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-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 
"#6. Compute the limit algebraically." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {OLE 1 4668 1 "[xm
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{XPPEDIT 343 0 "limit(x*sin(1/x),x = 0);" "6#-%&limitG6$*&%\"xG\"\"\"-
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