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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 10 "Calculus I" }}{PARA 258 "" 0 "" 
{TEXT -1 17 "Lesson 1:  Limits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We have seen that the limit " }
{XPPEDIT 18 0 "limit((f(x+h)-f(x))/h,h = 0);" "6#-%&limitG6$*&,&-%\"fG
6#,&%\"xG\"\"\"%\"hGF-F--F)6#F,!\"\"F-F.F1/F.\"\"!" }{TEXT -1 56 " giv
es the instantaneous rate of change of the function " }{XPPEDIT 18 0 "
f;" "6#%\"fG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT 
-1 38 ".  In computing this limit, the point " }{XPPEDIT 18 0 "x;" "6#
%\"xG" }{TEXT -1 64 " is fixed, we think of the difference quotient as
 a function of " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 33 ", and we \+
ask how it behaves when " }{XPPEDIT 18 0 "h;" "6#%\"hG" }{TEXT -1 51 "
 is close to 0.  Of course, we cannot actually put " }{XPPEDIT 18 0 "h
 = 0;" "6#/%\"hG\"\"!" }{TEXT -1 45 ", for then the quotient would be \+
undefined.  " }{TEXT 256 21 "Taking the limit as  " }{XPPEDIT 18 0 "pr
oc (h) options operator, arrow; 0 end;" "6#R6#%\"hG7\"6$%)operatorG%&a
rrowG6\"\"\"!F*F*F*" }{TEXT -1 1 " " }{TEXT 257 32 "is not the same as
 substituting " }{XPPEDIT 18 0 "h = 0;" "6#/%\"hG\"\"!" }{TEXT -1 161 
". In this worksheet, we will explore the idea of limit in general; in
 later worksheets, we will come back to the special type of limit abov
e.  We say a function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 42 ", \+
defined in some interval around a point " }{XPPEDIT 18 0 "a;" "6#%\"aG
" }{TEXT -1 25 ", but not necessarily at " }{XPPEDIT 18 0 "a;" "6#%\"a
G" }{TEXT -1 15 " itself, has a " }{TEXT 258 5 "limit" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "x;
" "6#%\"xG" }{TEXT -1 12 " approaches " }{XPPEDIT 18 0 "a;" "6#%\"aG" 
}{TEXT -1 9 ", written" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "limit(f(x),
x = a) = L;" "6#/-%&limitG6$-%\"fG6#%\"xG/F*%\"aG%\"LG" }{TEXT -1 1 ",
" }}{PARA 0 "" 0 "" {TEXT -1 15 "if we can make " }{XPPEDIT 18 0 "f(x)
;" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " as close to " }{XPPEDIT 18 0 "L;" 
"6#%\"LG" }{TEXT -1 22 " as we like by taking " }{XPPEDIT 18 0 "x;" "6
#%\"xG" }{TEXT -1 42 " sufficiently close to (but not equal to) " }
{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 94 ".  (This is a good enough d
efinition to get started; we will see a more precise one later on.)" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 267 "Lets lo
ok at limits from a geometric point of view. The concept of limits can
 seem difficult, but the idea is quite simple when you see what is hap
pening geometrically. We will construct a diagram which shows a functi
on and points approaching from the left and right." }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; w
ith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changeco
ords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "f := x -> 3 + (x-2)*cos((x-2));   a := 2:   left := -1:   right :=
5:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)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..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 = r
ed),  \n          plot(  [[ a+1/n, f(a + 1/n)] $n=1..20], x = left..ri
ght,  \n            style=point, symbol=circle, color = blue));" }}
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m@F-$\"3e([g_?dV;$F-7$$\"3zUr&G9dG9#F-$\"3L;&\\V*=STJF-7$$\"3+++++++D@
F-$\"3JmOS3Z-CJF-7$$\"3;6666666@F-$\"3;5KRXfU5JF-7$$\"33+++++++@F-$\"3
Q-y_;/]*4$F-7$$\"3#34444444#F-$\"3]'fa@pL04$F-7$$\"3[LLLLLL$3#F-$\"3I8
<*)[T/$3$F-7$$\"35xI#p2Bp2#F-$\"31tN01cpwIF-7$$\"3ir&G9dG92#F-$\"3'pjW
JVY72$F-7$$\"3)ommmmmm1#F-$\"3%[2!Qt&=l1$F-7$$\"3++++++]i?F-$\"31c(=%p
zPiIF-7$$\"3]k<THN#)e?F-$\"3]F>%zy@(eIF-7$$\"3Obbbbbbb?F-$\"3e_@sV)pa0
$F-7$$\"3eot%*y:j_?F-$\"3i4'\\))peD0$F-7$$\"3#)************\\?F-$\"3O[
(>I^P*\\IF--F\\[l6&F^[lF_[lF_[lF`[lFgclF[dl-%+AXESLABELSG6%Q\"x6\"Q!6
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 607 "
The green curve is the function. The yellow lines indicate where (a,f(
a)) is. The blue dots indicate points on f(x) as x approaches a from t
he right, and the green dots indicate points on f(x) where x is conver
ging to a from the left side. By looking at this diagram, you can gues
s the right limit by looking at what y value the points seem to be con
verging to. In a similar way, you can guess the left limit by looking \+
at what value the red dots seem to be converging to. If the red and bl
ue dots appear to be converging to the same value, then the limit exis
ts and equals the value they are converging to." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 172 "Here is another example \+
where the function is left and right limits are not the same. After re
-defining f(x), copy and paste the display command block above and re-
execute." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "f := x -> Heaviside(x-1) - Heaviside(1-x); a := 1:  l
eft := -2:  right :=4:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"
xG6\"6$%)operatorG%&arrowGF(,&-%*HeavisideG6#,&9$\"\"\"F2!\"\"F2-F.6#,
&F2F2F1F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 431 "displ
ay( 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, sym
bol=circle, color = red),  \n          plot(  [[ a+1/n, f(a + 1/n)] $n
=1..20], x = left..right,  \n            style=point, symbol=circle, c
olor = blue));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "
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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 \+
approaching from the right approach +1. In this case we say that the l
imit is undefined since there is not a single value for the limit." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "He
re are some more example functions.  For each of them, do the followin
g: " }}{PARA 259 "" 0 "" {TEXT -1 73 "(1) guess whether or not the fun
ction has a limit at the indicated point " }{XPPEDIT 18 0 "a;" "6#%\"a
G" }{TEXT -1 28 " and try evaluating f(a).  (" }{TEXT 265 34 "In most \+
cases, this will not work." }{TEXT -1 2 ") " }}{PARA 260 "" 0 "" 
{TEXT -1 102 "(2) check (or formulate) your guess numerically by evalu
ating the function at several points close to " }{XPPEDIT 18 0 "a;" "6
#%\"aG" }{TEXT -1 2 "; " }}{PARA 261 "" 0 "" {TEXT -1 82 "(3) graph th
e function and confirm that the graph has the expected behaviour near \+
" }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT -1 1 "." }}{PARA 262 
"" 0 "" {TEXT -1 47 "(4) use Maple to evaluate the limit of f(x) as " 
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