Théorème Unicité de la limite

Théorème):)Unicité)de)la)limite)
)
!"#$%f%&'(%)"'*$#"'%+,)#'#(%-&%."#/#'-0(%+(%-1%
Si%f%(/$%&'(%)"'*$#"'%2&#%-+3($%4-%4#3#$(%5%('%-%
Alors%f%'(%6(&$%6-/%/7-668"*9(8%+7&'(%-&$8(%4#3#$(1%%
%
:'%+7-&$8(/%$(83(/%;%%
Si%
lim
x→af(x)=L
%%($%/#%
lim
x→af(x)=M
%
Alors%
L=M
%%
%
<-66(4/%
=,)#'#$#"'%+(%4-%4#3#$(%;%
5%(/$%4-%4#3#$(%+(%f%('%-%/#%
∀ ε >0,
%%
∃ δ >0
%$(4%2&(%
∀x∈Df
%>%%
0<x−a< δ
%%"'%-%
f(x)−L< ε
%1%%
%
?',0-4#$,%+&%$8#-'04(%;%
!#%"'%-%&'%$8#-'04(%@AB%($%2&(%47"'%*-4*&4(8%4-%+#/$-'*(%+(%@%C%A>%4(%64&/%*"&8$%*9(3#'%(/$%
6-8%47-88D$(%+(%@%C%AE%('%6-//-'$%6-/%B>%4(%*9(3#'%(/$%64&/%4"'0%;%
A−B≤A−C+C−B
%%
%
%
=,3"'/$8-$#"'%
%
F"&8%+,3"'$8(8%2&(%f%'(%6(&$%6-/%-+3($$8(%+(&G%4#3#$(/%('%->%'"&/%-44"'/%/&66"/(8%2&(%f"
-+3($%+(&G%4#3#$(/%+#)),8('$(/%('%-%($%3"'$(8%2&(%*(4-%3H'(%C%&'(%*"'/,2&('*(%
-I/&8+(1%B(%$J6(%+(%+,3"'/$8-$#"'%/7-66(44(%démonstration"par"l’absurde1%
%
!&66"/"'/%+"'*%2&(%%
lim
x→af(x)=L
%%($%/#%
lim
x→af(x)=M
%-.(*%%
L≠M
%1%%
B9"#/#//"'/%-8I#$8-#8(3('$%
ε =L−M
4
>0
%%KLM%
!#%5%(/$%4-%4#3#$(%+(%f%2&-'+%G%$('+%.(8/%-%;%
∀ ε >0,
%%
∃ δ
L>0
%$(4%2&(%
∀x∈Df
%>%%
0<x−a< δ
L
%%"'%-%
f(x)−L< ε
%
!#%N%(/$%4-%4#3#$(%+(%f%2&-'+%G%$('+%.(8/%-%;%
∀ ε >0,
%%
∃ δ
M>0
%$(4%2&(%
∀x∈Df
%>%%
0<x−a< δ
M
%%"'%-%
f(x)−M< ε
%
B9"#/#//"'/%&'%x%/&))#/-33('$%68"*9(%+(%-%*7(/$OCO+#8(%C%&'(%+#/$-'*(%64&/%6($#$(%2&(%
δ
L
%
($%
δ
M
%+(%a1%%="'*%
f(x)−L< ε
%($%%
f(x)−M< ε
1%
P'%6(&$%,*8#8(%47#',0-4#$,%/&#.-'$(%;%%
4 ε =
(1)
L−M=L−f(x)+f(x)−M≤
Inégalité
du
triangle
L−f(x)
f(x)−L
   +f(x)−M<2
ε
%%
⇒4 ε <2 ε ⇒4 ε −2 ε <0⇒2 ε <0⇒ ε <0
%1%B(4-%(/$%-I/&8+(%6-8%9J6"$9H/(1%%
?4%(/$%+"'*%#36"//#I4(%+7-."#8%+(&G%4#3#$(/%+#)),8('$(/%('%-1%%
%
%%
%