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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Lab 18 Maple Activities fo
r December 5:  (Lesson 17 - Limits Involving Infinity)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Since we are working with
 limits, let's change the number of significant digits to 20:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "1.  In le
cture, we claimed that  " }{XPPEDIT 18 0 "Limit((5*x^2-3*x+1)/(2*x^2+4
*x-7),x = infinity);" "6#-%&LimitG6$*&,(*&\"\"&\"\"\"*$%\"xG\"\"#F*F**
&\"\"$F*F,F*!\"\"\"\"\"F*F*,(*&\"\"#F**$F,\"\"#F*F**&\"\"%F*F,F*F*\"\"
(F0F0/F,%)infinityG" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "5/2;" "6#*&\"
\"&\"\"\"\"\"#!\"\"" }{TEXT -1 46 " .  Let's see this using Maple.  Re
call that  " }{XPPEDIT 18 0 "Limit((5*x^2-3*x+1)/(2*x^2+4*x-7),x = inf
inity);" "6#-%&LimitG6$*&,(*&\"\"&\"\"\"*$%\"xG\"\"#F*F**&\"\"$F*F,F*!
\"\"\"\"\"F*F*,(*&\"\"#F**$F,\"\"#F*F**&\"\"%F*F,F*F*\"\"(F0F0/F,%)inf
inityG" }{TEXT -1 7 "    =  " }{XPPEDIT 18 0 "5/2;" "6#*&\"\"&\"\"\"\"
\"#!\"\"" }{TEXT -1 11 "  means as " }{TEXT 259 1 "x" }{TEXT -1 34 " a
pproaches infinity (that is, as " }{TEXT 265 1 "x" }{TEXT -1 16 " gets
 \"large\"), " }{XPPEDIT 18 0 "(5*x^2-3*x+1)/(2*x^2+4*x-7);" "6#*&,(*&
\"\"&\"\"\"*$%\"xG\"\"#F'F'*&\"\"$F'F)F'!\"\"\"\"\"F'F',(*&\"\"#F'*$F)
\"\"#F'F'*&\"\"%F'F)F'F'\"\"(F-F-" }{TEXT -1 12 " approaches " }
{XPPEDIT 18 0 "5/2" "6#*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 15 ".  Let's
 let f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 ") =  " }{XPPEDIT 
18 0 "(5*x^2-3*x+1)/(2*x^2+4*x-7);" "6#*&,(*&\"\"&\"\"\"*$%\"xG\"\"#F'
F'*&\"\"$F'F)F'!\"\"\"\"\"F'F',(*&\"\"#F'*$F)\"\"#F'F'*&\"\"%F'F)F'F'
\"\"(F-F-" }{TEXT -1 14 "  and show as " }{XPPEDIT 18 0 "x;" "6#%\"xG
" }{TEXT -1 17 " gets \"large\", f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 13 ") approaches " }{XPPEDIT 18 0 "5/2" "6#*&\"\"&\"\"\"\"\"#
!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f \+
:= x -> (5*x^2 - 3*x + 1)/(2*x^2 + 4*x - 7);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Let's let " }{TEXT 260 1 "x" }{TEXT -1 
28 " get \"large\".  That is, let " }{TEXT 264 1 "x" }{TEXT -1 19 " ap
proach infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f(1000.
); f(100000.); f(10000000.); f(10000000000.); f(100000000000000.);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "It does appear that as " }{TEXT 
261 1 "x" }{TEXT -1 27 " gets \"large\" (that is, as " }{TEXT 262 1 "x
" }{TEXT -1 25 " approaches infinity), f(" }{TEXT 263 1 "x" }{TEXT -1 
23 ") is approaching 2.5 = " }{XPPEDIT 18 0 "5/2" "6#*&\"\"&\"\"\"\"\"
#!\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 67 "Now, let's see the Maple command, that wi
ll find this limit for us:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "Limit(f(x), x=infinity); value(%);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "2.  In le
cture, we claimed that  " }{XPPEDIT 18 0 "Limit((2*x^2-x+3)/(x^3-1),x \+
= infinity);" "6#-%&LimitG6$*&,(*&\"\"#\"\"\"*$%\"xG\"\"#F*F*F,!\"\"\"
\"$F*F*,&*$F,\"\"$F*\"\"\"F.F./F,%)infinityG" }{TEXT -1 50 "  =  0.  L
et's see this using Maple.  Recall that " }{XPPEDIT 18 0 "Limit((2*x^2
-x+3)/(x^3-1),x = infinity);" "6#-%&LimitG6$*&,(*&\"\"#\"\"\"*$%\"xG\"
\"#F*F*F,!\"\"\"\"$F*F*,&*$F,\"\"$F*\"\"\"F.F./F,%)infinityG" }{TEXT 
-1 19 "    =  0  means as " }{TEXT 266 1 "x" }{TEXT -1 34 " approaches
 infinity (that is, as " }{TEXT 272 1 "x" }{TEXT -1 16 " gets \"large
\"), " }{XPPEDIT 18 0 "(2*x^2-x+3)/(x^3-1);" "6#*&,(*&\"\"#\"\"\"*$%\"
xG\"\"#F'F'F)!\"\"\"\"$F'F',&*$F)\"\"$F'\"\"\"F+F+" }{TEXT -1 28 " app
roaches 0.  Let's let f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 ")
 =  " }{XPPEDIT 18 0 "(2*x^2-x+3)/(x^3-1);" "6#*&,(*&\"\"#\"\"\"*$%\"x
G\"\"#F'F'F)!\"\"\"\"$F'F',&*$F)\"\"$F'\"\"\"F+F+" }{TEXT -1 14 "  and
 show as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 17 " gets \"large\"
, f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 15 ") approaches 0." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f := x -> (2*x^2 - x + 3)/(x
^3 - 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Le
t's let " }{TEXT 267 1 "x" }{TEXT -1 28 " get \"large\".  That is, let
 " }{TEXT 271 1 "x" }{TEXT -1 19 " approach infinity." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f(1000.); f(100000.); f(10000000.);
 f(10000000000.); f(100000000000000.);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "It does appear that as " 
}{TEXT 268 1 "x" }{TEXT -1 27 " gets \"large\" (that is, as " }{TEXT 
269 1 "x" }{TEXT -1 25 " approaches infinity), f(" }{TEXT 270 1 "x" }
{TEXT -1 19 ") is approaching 0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Now, let's see the Maple command, \+
that will find this limit for us:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "Limit(f(x), x=infinity); value(%);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 80:" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "3
.  In lecture, we claimed that  " }{XPPEDIT 18 0 "Limit(cos(1/(x^3)),x
 = -infinity);" "6#-%&LimitG6$-%$cosG6#*&\"\"\"F**$)%\"xG\"\"$F*!\"\"/
%\"xG,$%)infinityG!\"\"" }{TEXT -1 50 "  =  1.  Let's see this using M
aple.  Recall that " }{XPPEDIT 18 0 "Limit(cos(1/(x^3)),x = -infinity)
;" "6#-%&LimitG6$-%$cosG6#*&\"\"\"F**$)%\"xG\"\"$F*!\"\"/%\"xG,$%)infi
nityG!\"\"" }{TEXT -1 17 "  =  1  means as " }{TEXT 280 1 "x" }{TEXT 
-1 43 " approaches negative infinity (that is, as " }{TEXT 262 1 "x" }
{TEXT -1 27 " gets \"large\" negatively), " }{XPPEDIT 18 0 "cos(1/(x^3
));" "6#-%$cosG6#*&\"\"\"F'*$)%\"xG\"\"$F'!\"\"" }{TEXT -1 28 " approa
ches 1.  Let's let f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 ") = \+
 " }{XPPEDIT 18 0 "cos(1/(x^3));" "6#-%$cosG6#*&\"\"\"F'*$)%\"xG\"\"$F
'!\"\"" }{TEXT -1 14 "  and show as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }
{TEXT -1 28 " gets \"large\" negatively, f(" }{XPPEDIT 18 0 "x;" "6#%
\"xG" }{TEXT -1 15 ") approaches 1." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "f := x -> cos(1/x^3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 10 "Let's let " }{TEXT 281 1 "x" }{TEXT -1 
39 " get \"large\" negatively.  That is, let " }{TEXT 261 1 "x" }
{TEXT -1 28 " approach negative infinity." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 78 "f(-1000.); f(-100000.); f(-10000000.); f(-100000000
00.); f(-100000000000000.);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 23 "It does appear that as " }{TEXT 282 1 "x
" }{TEXT -1 38 " gets \"large\" negatively (that is, as " }{TEXT 259 
1 "x" }{TEXT -1 34 " approaches negative infinity), f(" }{TEXT 260 1 "
x" }{TEXT -1 19 ") is approaching 1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Now, let's see the Maple comman
d, that will find this limit for us:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "Limit(f(x), x=-infinity); value(%);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 20:" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "4
.  In lecture, we claimed that  " }{XPPEDIT 18 0 "Limit((3-5*x)/(sqrt(
4*x^2+9*x)),x = -infinity);" "6#-%&LimitG6$*&,&\"\"$\"\"\"%\"xG!\"&\"
\"\"*$-%%sqrtG6#,&*$)F*\"\"#F,\"\"%F*\"\"*F,!\"\"/F*,$%)infinityG!\"\"
" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "5/2;" "6#*&\"\"&\"\"\"\"\"#!\"\"
" }{TEXT -1 44 ".  Let's see this using Maple.  Recall that " }
{XPPEDIT 18 0 "Limit((3-5*x)/(sqrt(4*x^2+9*x)),x = -infinity);" "6#-%&
LimitG6$*&,&\"\"$\"\"\"%\"xG!\"&\"\"\"*$-%%sqrtG6#,&*$)F*\"\"#F,\"\"%F
*\"\"*F,!\"\"/F*,$%)infinityG!\"\"" }{TEXT -1 5 "  =  " }{XPPEDIT 18 
0 "5/2;" "6#*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 11 "  means as " }
{TEXT 260 1 "x" }{TEXT -1 43 " approaches negative infinity (that is, \+
as " }{TEXT 259 1 "x" }{TEXT -1 27 " gets \"large\" negatively), " }
{XPPEDIT 18 0 "(3-5*x)/(sqrt(4*x^2+9*x));" "6#*&,&\"\"$\"\"\"%\"xG!\"&
\"\"\"*$-%%sqrtG6#,&*$)F'\"\"#F)\"\"%F'\"\"*F)!\"\"" }{TEXT -1 12 " ap
proaches " }{XPPEDIT 18 0 "5/2;" "6#*&\"\"&\"\"\"\"\"#!\"\"" }{TEXT 
-1 15 ".  Let's let f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 ") =
  " }{XPPEDIT 18 0 "(3-5*x)/(sqrt(4*x^2+9*x));" "6#*&,&\"\"$\"\"\"%\"x
G!\"&\"\"\"*$-%%sqrtG6#,&*$)F'\"\"#F)\"\"%F'\"\"*F)!\"\"" }{TEXT -1 
14 "  and show as " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 28 " gets \+
\"large\" negatively, f(" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 13 "
) approaches " }{XPPEDIT 18 0 "5/2;" "6#*&\"\"&\"\"\"\"\"#!\"\"" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := x -> \+
(3-5*x)/sqrt(4*x^2+9*x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "
f(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 10 "Let's let " }{TEXT 261 1 "x" }{TEXT -1 39 " get \"large\"
 negatively.  That is, let " }{TEXT 306 1 "x" }{TEXT -1 28 " approach \+
negative infinity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f(-10
00.); f(-100000.); f(-10000000.); f(-10000000000.); f(-100000000000000
.);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 23 "It does appear that as " }{TEXT 262 1 "x" }{TEXT -1 38 " gets \+
\"large\" negatively (that is, as " }{TEXT 304 1 "x" }{TEXT -1 34 " ap
proaches negative infinity), f(" }{TEXT 305 1 "x" }{TEXT -1 23 ") is a
pproaching 2.5 = " }{XPPEDIT 18 0 "5/2" "6#*&\"\"&\"\"\"\"\"#!\"\"" }
{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 67 "Now, let's see the Maple command, that will find thi
s limit for us:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit(f(
x), x=-infinity); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 307 0 "" }{TEXT 308 7 "Example" 
}{TEXT 309 0 "" }{TEXT -1 7 "  Find " }{XPPEDIT 18 0 "Limit((x-1)/(x+2
),x = -2,right);" "6#-%&LimitG6%*&,&%\"xG\"\"\"!\"\"F)\"\"\",&F(F)\"\"
#F)!\"\"/F(!\"#%&rightG" }{TEXT -1 3 "  ." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> (x-1)/(x+2
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " 
}{TEXT 310 1 "x" }{TEXT -1 29 " approach - 2 from the right:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f(-1.9999); f(-1.99999999); \+
f(-1.999999999999);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 16 "It appears that " }{TEXT 311 1 "f" }{TEXT -1 1 
"(" }{TEXT 312 1 "x" }{TEXT -1 19 ") is approaching - " }{XPPEDIT 18 
0 "infinity;" "6#%)infinityG" }{TEXT -1 41 ".  Now, let's find the lim
it using Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Limit(f(
x), x=-2, right); value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 313 0 "" }{TEXT 314 7 "Example" 
}{TEXT 315 0 "" }{TEXT -1 7 "  Find " }{XPPEDIT 18 0 "Limit((x-1)/(x+2
),x = -2,left);" "6#-%&LimitG6%*&,&%\"xG\"\"\"!\"\"F)\"\"\",&F(F)\"\"#
F)!\"\"/F(!\"#%%leftG" }{TEXT -1 3 "  ." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := x -> (x-1)/(x+2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }
{TEXT 321 1 "x" }{TEXT -1 28 " approach - 2 from the left:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "f(-2.0001); f(-2.00000001); f(-2.00
0000000001);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 16 "It appears that " }{TEXT 319 1 "f" }{TEXT -1 1 "(" }
{TEXT 320 1 "x" }{TEXT -1 17 ") is approaching " }{XPPEDIT 18 0 "infin
ity;" "6#%)infinityG" }{TEXT -1 41 ".  Now, let's find the limit using
 Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "Limit(f(x), x=-2, left); value(%);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
322 0 "" }{TEXT 323 7 "Example" }{TEXT 324 0 "" }{TEXT -1 7 "  Find " 
}{XPPEDIT 18 0 "limit((-3)/((x-1)^2),x = 1);" "6#-%&limitG6$*&,$\"\"$!
\"\"\"\"\"*$),&%\"xG\"\"\"!\"\"F/\"\"#\"\"\"!\"\"/%\"xG\"\"\"" }{TEXT 
-1 14 "  using Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Limit(-3/(x-1)^2,x=1); value(%);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }{TEXT 325 5 "NOTE:" }{TEXT -1 69 "  This answer means that both the
 right-hand and left-hand limits are" }{TEXT -1 3 " - " }{XPPEDIT 18 
0 "infinity;" "6#%)infinityG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Practice Problems:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Use Maple
 to find the following limits.  However, only use the " }{TEXT 256 6 "
Limit " }{TEXT -1 180 "command to check your answer.  You might want t
o work with twenty significant digits instead of the default value of \+
ten.  To get twenty significant digits, type the Maple command " }
{TEXT 257 15 "Digits := 20:  " }{TEXT -1 59 "You will need to retype t
his command after each use of the " }{TEXT 258 7 "restart" }{TEXT -1 
9 " command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "1.   " }{XPPEDIT 18 0 "Limit(x/(2-3*x),x = -infinity);" "6#-%&Li
mitG6$*&%\"xG\"\"\",&\"\"#F(*&\"\"$F(F'F(!\"\"F-/F',$%)infinityGF-" }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 "2.  " }{XPPEDIT 18 0 "Limit(sqrt((2*x-1)/(
x^2+1)),x = infinity);" "6#-%&LimitG6$-%%sqrtG6#*&,&*&\"\"#\"\"\"%\"xG
F-F-\"\"\"!\"\"F-,&*$F.\"\"#F-\"\"\"F-F0/F.%)infinityG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "3.  " }{XPPEDIT 18 0 "Limi
t((4*x^2-3)/((2*x^6-5)^(1/3)),x = -infinity);" "6#-%&LimitG6$*&,&*&\"
\"%\"\"\"*$%\"xG\"\"#F*F*\"\"$!\"\"F*),&*&\"\"#F**$F,\"\"'F*F*\"\"&F/*
&\"\"\"F*\"\"$F/F//F,,$%)infinityGF/" }}}{EXCHG {PARA 0 "> " 0 "" 
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