Menu Close

find-lim-x-0-sinx-cosx-1-x-




Question Number 30502 by abdo imad last updated on 22/Feb/18
find  lim_(x→0) (sinx +cosx)^(1/x)   .
findlimx0(sinx+cosx)1x.
Answered by sma3l2996 last updated on 24/Feb/18
(sinx+cosx)^(1/x) =e^((ln(sinx+cosx))/x)   sin(x)∼_0 x−(x^3 /(3!))  and  cos(x)∼_0 1−(x^2 /2)  so  sinx+cosx∼_0 1+x−(x^2 /2)−(x^3 /6)  ln(cosx+sinx)∼_0 ln(1+x−x^2 /2−x^3 /6)∼_0 x−x^2 /2−x^3 /6  ((ln(sinx+cosx))/x)∼_0 1  so lim_(x→0) (sinx+cosx)^(1/x) =e
(sinx+cosx)1/x=eln(sinx+cosx)xsin(x)0xx33!andcos(x)01x22sosinx+cosx01+xx22x36ln(cosx+sinx)0ln(1+xx2/2x3/6)0xx2/2x3/6ln(sinx+cosx)x01solimx0(sinx+cosx)1/x=e

Leave a Reply

Your email address will not be published.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