Menu Close

lim-x-sin-x-1-x-sin-x-




Question Number 119517 by bemath last updated on 25/Oct/20
 lim_(x→∞)  [ sin (x+(1/x))−sin x ] =?
limx[sin(x+1x)sinx]=?
Answered by Dwaipayan Shikari last updated on 25/Oct/20
lim_(x→∞) (sin(x+(1/x))−sinx)  =lim_(x→∞) (2cos(x+(1/(2x)))sin((1/x)))=0  −1≤cos(x+(1/(2x)))≤1  lim_(x→∞) sin(1/x)→0
limx(sin(x+1x)sinx)=limx(2cos(x+12x)sin(1x))=01cos(x+12x)1limxsin1x0
Answered by benjo_mathlover last updated on 25/Oct/20
 lim_(x→∞)  [ sin xcos (1/x)+cos xsin (1/x)−sin x ] =   lim_(x→∞)  [ sin x(cos (1/x)−1)+cos x sin (1/x) ]  [ note lim_(x→∞)  cos (1/x)−1 = 0 ∧ lim_(x→∞)  cos xsin (1/x) = 0   because cos x oscilates between −1 and 1 ]  thus lim_(x→∞)  [ sin x(cos (1/x)−1)+cos xsin (1/x) ] = 0
limx[sinxcos1x+cosxsin1xsinx]=limx[sinx(cos1x1)+cosxsin1x][notelimxcos1x1=0limxcosxsin1x=0becausecosxoscilatesbetween1and1]thuslimx[sinx(cos1x1)+cosxsin1x]=0

Leave a Reply

Your email address will not be published. Required fields are marked *