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lim-x-0-x-2-cos-x-3sin-x-x-5-




Question Number 131836 by liberty last updated on 09/Feb/21
lim_(x→0)  ((x(2+cos x)−3sin x)/x^5 )
limx0x(2+cosx)3sinxx5
Commented by EDWIN88 last updated on 10/Feb/21
another way L′Ho^� pital   lim_(x→0) ((2x+xcos x−3sin x)/x^5 )=   lim_(x→0)  ((2+cos x−xsin x−3cos x)/(5x^4 ))=lim_(x→0) ((2−2cos x−xsin x)/(5x^4 ))   lim_(x→0) ((2sin x−(sin x+xcos x))/(20x^3 ))=lim_(x→0) ((sin x−xcos x)/(20x^3 ))   =lim_(x→0) ((cos x−(cos x−xsin x))/(60x^2 ))  =lim_(x→0) ((xsin x)/(60x^2 )) = (1/(60))
anotherwayLHopital^limx02x+xcosx3sinxx5=limx02+cosxxsinx3cosx5x4=limx022cosxxsinx5x4limx02sinx(sinx+xcosx)20x3=limx0sinxxcosx20x3=limx0cosx(cosxxsinx)60x2=limx0xsinx60x2=160
Answered by EDWIN88 last updated on 10/Feb/21
 lim_(x→0)  ((x(2+1−(x^2 /2)+(x^4 /(24)))−3(x−(x^3 /6)+(x^5 /(120))))/x^5 ) =   lim_(x→0)  ((3x−(x^3 /2)+(x^5 /(24))−3x+(x^3 /2)−(x^5 /(40)))/x^5 ) =   lim_(x→0)  ((x^5 ((1/(8.3))−(1/(8.5))))/x^5 ) = ((5−3)/(8.5.3)) = (1/(4.5.3))=(1/(60))
limx0x(2+1x22+x424)3(xx36+x5120)x5=limx03xx32+x5243x+x32x540x5=limx0x5(18.318.5)x5=538.5.3=14.5.3=160
Commented by malwan last updated on 10/Feb/21
thank you sir
thankyousir
Answered by malwan last updated on 09/Feb/21
lim_(x→0)  ((x(2+1−(x^2 /(2!))+(x^4 /(4!))∓..)−3(x−(x^3 /(3!))+(x^5 /(5!))∓..))/x^5 )  =lim_(x→0)  (((3x−(x^3 /2)+(x^5 /(24))∓..)−(3x−(x^3 /2)+(x^5 /(40))∓..))/x^5 )  =lim_(x→0)  (((((5−3)/(120)))x^5 +..)/x^5 ) = (2/(120)) = (1/(60))
limx0x(2+1x22!+x44!..)3(xx33!+x55!..)x5=limx0(3xx32+x524..)(3xx32+x540..)x5=limx0(53120)x5+..x5=2120=160

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