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=.&4=)R%*fF17$Fhp$\"3*=\"yD#)oEgeF17$F]q$\"36ilVV6g6dF17$Fbq$\"3SZ#p')
*=;'e&F17$Fgq$\"3Ohosc[saaF17$F\\r$\"3%)G,(f\"))=N`F17$Far$\"3*e#\\#fo
Df@&F17$Ffr$\"36pr$*4p+6^F17$F[s$\"3z9;nm\"pC+&F17$F`s$\"3kAqJJY]%*[F1
7$Fes$\"3.#GB#*HXU![F17$Fjs$\"3[V_oM*G/r%F17$F_t$\"3$3b6mKzsh%F17$Fdt$
\"3#4W&oC!['HXF17$Fit$\"3mzJ\\*HpzW%F17$F^u$\"36s7$fMi1O%F17$Fcu$\"3jT
\"=8&f3&G%F17$Fhu$\"3S%)fbc7B2UF17$F]v$\"3$4f8%**p3RTF17$Fbv$\"3S`,**
\\#pq1%F17$Fgv$\"3@G9s@xb,SF17$F\\w$\"3&z)zhpzHNRF17$Faw$\"3U6K'pc/E(Q
F17$Ffw$\"38)4`Hcv!4QF17$F[x$\"3lJs8#RH)\\PF17$F`x$\"3qa))>zy6\"p$F17$
Fex$\"3WL2)y\\!oMOF17$Fjx$\"33'pj+!4K%e$F17$F_y$\"3/6U;JDHGNF17$Fdy$\"
3Lj#=^eV'zMF17$Fiy$\"3#)ee1%**H#HMF17$F^z$\"3#[PZ6CFBQ$F17$Fcz$\"3:LLL
LLLLLF1-Fhz6&FjzF^[lF[[lF^[l-%+AXESLABELSG6$Q\"x6\"Q!Fhdl-%%VIEWG6$;F(
Fcz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 -61.000000 -138.000000 
0 0 "Curve 1" "Curve 2" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exer
cice 15" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit((1-2/n)^(3*
n),n=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#!\"'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Incroyable !" }}}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 11 "Exercice 16" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "taylor(1/sin(x)-1/x,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+
)%\"xG#\"\"\"\"\"'F&#\"\"(\"$g$\"\"$-%\"OG6#F&\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 104 "C'est un \"d\351veloppement de Taylor\" (th
\351orie en cours d'ann\351e). Le premier terme nous donne l'\351quiva
lent." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 17" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "taylor(1/sin(x)^2-1/x^2,x=0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"xG#\"\"\"\"\"$\"\"!#F&\"#:\"\"##F
+\"$*=\"\"%-%\"OG6#F&\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "limit(1/sin(x)^2-1/x^2,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##
\"\"\"\"\"$" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 20" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "limit((t^t-1)/(1-t+ln(1+t)),
t=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "limit(ln(t)/(t-1),t=1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "li
mit((1-1/(n*sqrt(n)))^(n^(5/3)),n=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "un peu p
lus subtil" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit((t^t-1)/
(1-t+ln(t)),t=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "limit((t^t-1)/(1-t+ln(t)),t=
1,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "limit((t^t-1)/(1-t+ln(t)),t=
1,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "taylor((t^t-1)/(1-t+ln(t)),t=1);" }
}{PARA 8 "" 1 "" {TEXT -1 54 "Error, does not have a taylor expansion,
 try series()\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "series((
t^t-1)/(1-t+ln(t)),t=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-,&%\"tG
\"\"\"F&!\"\"!\"#F'#!#5\"\"$\"\"!#!#?\"\"*F&#!$t\"\"$N\"\"\"#-%\"OG6#F
&F+" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "limit(x^(sqrt(x))/(
sqrt(x))^x,x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "limit((Pi/2-x)^(sin(x)/ln(co
s(x))),x=Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit(tan(x)^(tan(2*x)),x=0
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 11 "Exercice 21" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "limit(1/(1-x^x)-1/(x*ln(x)),x=0,right);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "limit(sqrt(x+sqrt(x))-sqrt(x),x=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "limit(cos(x)^(1/cotan(x)^2),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "limit(cos(x)^(
1/tan(x)^2),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6##!\"\"\"
\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "?????" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 12 "cotan(Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%&cotanG6#,$%#PiG#\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
145 "Il n connait pas cette fonction... cela dit, il est \"assez conte
stable\" de donner la  r\351ponse qu'il donne sur limit(cos(x)^(1/cota
n(x)^2),x=0)..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "limit(ta
n(x)^(tan(2*x)),x=Pi/2,left);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit(tan(x)^(tan(2*x
)),x=Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#!\"\"" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice 22" }}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 4 "22.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "l
imit(1/cos(x)^2+1/ln(sin(x)^2),x=Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6##\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "taylo
r(ln(sin(x)^2)+cos(x)^2,x=Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'
,&%\"xG\"\"\"*&#F&\"\"#F&%#PiGF&!\"\"#F+F)\"\"%-%\"OG6#F&\"\"'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Soit encore :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 44 "taylor(ln(sin(Pi/2+u)^2)+cos(Pi/2+u)^2,u=0);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"uG#!\"\"\"\"#\"\"%-%\"OG6#\"
\"\"\"\"'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 4 "22.2" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "limit((x^sin(x)-sin(x)^x)/(x^tan(x)
-tan(x)^x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!\"\"\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "taylor(x^sin(x)-sin(x)^x,x=0
);" }}{PARA 8 "" 1 "" {TEXT -1 54 "Error, does not have a taylor expan
sion, try series()\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ser
ies(x^sin(x)-sin(x)^x,x=0,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"
xG,&#\"\"\"\"\"'F'*&#F'F(F'-%#lnG6#F$F'!\"\"\"\"$-%\"OG6#F'\"\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "series(x^tan(x)-tan(x)^x,x=0
,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"xG,&#!\"\"\"\"$\"\"\"*&#F
)F(F)-%#lnG6#F$F)F)F(-%\"OG6#F)\"\"%" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 4 "22.3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "limit((2
^x+3^x-12)^(tan(Pi/4*x)),x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"
\"\"F$*&))\"\"#*&F$F$%#PiG!\"\"\"#;F$))\"\"$F)\"#OF$F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#)\"7cQRMT#)=?gO)*,$*&\"\"\"F'%#PiG!\"\"F)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "mouais..." }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 4 "22.4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "limit((e
xp(1)-(1+1/x)^x)^(sqrt(x^2+1)-sqrt(x^2-1)),x=infinity);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "taylor(exp(1)-(1+1/x)^x,x=infinity,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&-%$expG6#\"\"\"F(%\"xG!\"\"#F(\"\"#-%\"OG6#*&F(F(*$
)F)F,F(F*F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "taylor(sqrt(
x^2+1)-sqrt(x^2-1),x=infinity,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
&*&\"\"\"F%%\"xG!\"\"F%-%\"OG6#*&F%F%*$)F&\"\"$F%F'F%" }}}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 11 "Exercice 23" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "series(ln(tan(x)),x=0,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+'%\"xG-%#lnG6#F$\"\"!-%\"OG6#\"\"\"\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "taylor(ln(tan(x)),x=Pi/4,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+',&%\"xG\"\"\"*&#F&\"\"%F&%#PiGF&!\"
\"\"\"#F&-%\"OG6#F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 
"series(sqrt(x^2+x)-(x^3+2*x^2)^(1/3),x=0,1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*$-%%sqrtG6#%\"xG\"\"\"F)-%\"OG6#*$)F(#\"\"#\"\"$F)F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "asympt(sqrt(x^2+x)-(x^3
+2*x^2)^(1/3),x,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#!\"\"\"\"'\"
\"\"*&#\"#B\"#sF'%\"xGF%F'-%\"OG6#*&F'F'*$)F,\"\"#F'F%F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "series(1/x-1/tan(x),x=0,4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"xG#\"\"\"\"\"$F&-%\"OG6#F&\"\"#" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "asympt(exp(1/x)-x*(x+1)/x
^2,x,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*$)%\"xG\"\"#F
%!\"\"#F%F)-%\"OG6#*&F%F%*$)F(\"\"$F%F*F%" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 42 "asympt(sqrt(ln(2*n+1))-sqrt(ln(2*n)),n,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*&-%%sqrtG6#,&-%#lnG6#\"\"
#F%-F,6#%\"nGF%F%F1F%!\"\"#F%\"\"%-%\"OG6#*&F%F%*$)F1F.F%F2F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "asympt((ln(n+1)/ln(n))^n-1,n
,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#*&\"\"\"F(-%#lnG6#%
\"nG!\"\"F(F(F--%\"OG6#*&F(F(F,F-F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$
expG6#*&\"\"\"F(-%#lnG6#%\"nG!\"\"F(F(F--%\"OG6#*&F(F(F,F-F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Etrange... je n'ai pas r\351ussi \+
\340 lui faire \"voir\" que exp(1/ln(n))-1 est \351quivalent \340 1/ln
(n). Comme quoi..." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Exercice \+
24" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "asympt((ln(x)/ln(x-1))
^(x^2),x,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#,$*&,&-%#
lnG6#%\"xG!\"\"\"\"\"F/F0F+!\"##F/\"\"#F0-%\"OG6#*&F0F0F.F/F0F0-F&6#,$
*&F+F/F.F0F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Arf ! On fait l
e calcul \340 la main... et onpeut tout de m\352me v\351rifier :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "limit(exp(-x/ln(x))*(ln(x)/l
n(x-1))^(x^2),x=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "asympt(exp(sqrt(x^2+x+1))
,x,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6##\"\"\"\"\"#F)-
%\"OG6#*&F)F)%\"xG!\"\"F)F)-F&6#F/F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$
expG6#%\"xG\"\"\"-F&6##F)\"\"#F)F)*&F%F)-%\"OG6#*&F)F)F(!\"\"F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "O(1/x) est uu terme qui tend vers \+
0, donc le tout est \351quivalent \340 exp(x)exp(1/2)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "series(exp(tan(x)^2),x=Pi/2);" }}
{PARA 8 "" 1 "" {TEXT -1 48 "Error, (in series/exp) unable to compute \+
series\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "series(tan(x)^2
,x=Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-,&%\"xG\"\"\"*&#F&\"\"#
F&%#PiGF&!\"\"F&!\"##F,\"\"$\"\"!#F&\"#:F)#F)\"$*=\"\"%-%\"OG6#F&\"\"&
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "\"donc\" exp(tan^2(Pi/2+u)) e
st equivalent \340 exp(-2/3)exp(1/u^2)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "asympt(ln(n)/n-ln(n)^2/2+ln(n-1)^2/2,n,3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*&,&-%#lnG6#%\"nG#!\"\"\"\"##\"\"\"F,F.F.F
)!\"#F.-%\"OG6#*&F.F.*$)F)\"\"$F.F+F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "asympt((ln(x)/ln(x-1))^(x^2),x,2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#*&,&-%$expG6#,$*&,&-%#lnG6#%\"xG!\"\"\"\"\"F/F0F+!\"#
#F/\"\"#F0-%\"OG6#*&F0F0F.F/F0F0-F&6#,$*&F+F/F.F0F/F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&,&-%$expG6#,$*&,&-%#lnG6#%\"xG\"\"\"F/F/F/F+!\"##F/\"
\"#F/-%\"OG6#*&F/F/F.!\"\"F/F/-F&6#*&F+F7F.F/F/" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 11 "A la main :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)inf
inityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "asympt(ln(x)/ln(x
-1),x,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*\"\"\"F$*&F$F$*&-%#lnG6
#%\"xGF$F*F$!\"\"F$*(,&#F$\"\"#F$*&F$F$F'F+F$F$F'F+F*!\"#F$-%\"OG6#*&F
$F$*$)F*\"\"$F$F+F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "asym
pt(x^2*ln(ln(x)/ln(x-1)),x,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&
-%#lnG6#%\"xG!\"\"F(\"\"\"F**&,&#F*\"\"#F**&F*F*F%F)F*F*F%F)F**&#F*F.F
**&F*F**$)F%F.F*F)F*F)-%\"OG6#*&F*F*F(F)F*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 10 "Et donc..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "
series(x^x-(sin(x))^x-x^3/6,x=0,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#+'%\"xG,$-%#lnG6#F$#\"\"\"\"\"'\"\"%-%\"OG6#F*\"\"&" }}}}}{MARK "19 2
 0 0" 70 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 1 33 1 1 
}