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Question Number 134618 by Abdoulaye last updated on 05/Mar/21
  prove that cos (x) does not admit a  limt in +∞?
provethatcos(x)doesnotadmitalimtin+?
Answered by Olaf last updated on 05/Mar/21
Let x_n  = cosn  Let y_n  = cos(n−1)  Let z_n  = cos(n+1)  Let t_n  = cos(2n)  Suppose now that x_n  admits a limit l  in +∞ :  lim_(n→∞)  x_n  = l  In this case we have too :  lim_(n→∞)  y_n  = lim_(n→∞)  z_n  = lim_(n→∞)  t_n  = l  But y_n +z_n  = 2x_n cos(1)  ⇒ l+l = 2l.cos(1) ⇒ l = 0 (1)  And t_n  = 2x_n ^2 −1  ⇒ l = 2l^2 −1 ⇒ l = −(1/2) or +1 (2)    (1) and (2) are not compatible !
Letxn=cosnLetyn=cos(n1)Letzn=cos(n+1)Lettn=cos(2n)Supposenowthatxnadmitsalimitlin+:limnxn=lInthiscasewehavetoo:limnyn=limnzn=limntn=lButyn+zn=2xncos(1)l+l=2l.cos(1)l=0(1)Andtn=2xn21l=2l21l=12or+1(2)(1)and(2)arenotcompatible!
Answered by mathmax by abdo last updated on 06/Mar/21
if this limit exist for all sequence u_n /lim u_n =+∞  lim cos(u_n ) are the same  but for u_n =nπ  cos(u_n )=(−1)^n  and this sequence is not convergent...
ifthislimitexistforallsequenceun/limun=+limcos(un)arethesamebutforun=nπcos(un)=(1)nandthissequenceisnotconvergent
Commented by Abdoulaye last updated on 06/Mar/21
thank you sir
thankyousir
Commented by mathmax by abdo last updated on 06/Mar/21
you are welcome sir
youarewelcomesir

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