Question Number 149959 by Mathfinity last updated on 08/Aug/21

Commented byjohn_santu last updated on 08/Aug/21

Commented byjohn_santu last updated on 08/Aug/21

Commented byjohn_santu last updated on 08/Aug/21

Answered by gsk2684 last updated on 08/Aug/21

i) lim_(x→0) (((2x−(((2x)^3 )/(3!))+..)+(6x−(((6x)^3 )/(3!))+..)+(10x−(((10x)^3 )/(3!))+..)−(18x−(((18x)^3 )/(3!))+..))/(3(x−(x^3 /(3!))+..)−(3x−(((3x)^3 )/(3!))+..))) lim_(x→0) ((((−8−216−1000+5832)/6)x^3 +..)/(((−3+27)/6)x^3 +..)) =(((−8−216−1000+5832)/6)/((−3+27)/6)) =((4608)/(24))=192 or lim_(x→0) ((2 sin ((6x+2x)/2) cos((6x−2x)/2) +2 cos ((10x+18x)/2) sin ((10x−18x)/2))/(4 sin ^3 x)) =lim_(x→0) ((2 sin 4x. cos 2x −2 cos 14x. sin 4x)/(4 sin ^3 x)) =lim_(x→0) ((2 sin 4x( cos 2x − cos 14x))/(4 sin ^3 x)) =lim_(x→0) ((2 sin 4x.2 sin ((14x+2x)/2) sin ((14x−2x)/2))/(4 sin ^3 x)) =lim_(x→0) (( sin 4x. sin 8x. sin 6x)/( sin ^3 x)) =4×8×6=192 ii)lim_(x→0) ((1−cos ((π/2)x+sin x))/(2x^2 )) =lim_(x→0) ((2sin ^2 ((π/4)x+((sin x)/2)))/(2x^2 )) =lim_(((π/4)x+((sin x)/2))→0) ((sin ^2 ((π/4)x+((sin x)/2)))/(((π/4)x+((sin x)/2))^2 ))lim_(x→0) ((((π/4)x+((sin x)/2))^2 )/x^2 ) =(1)^2 lim_(x→0) ((π/4)+(1/2)((sin x)/x))^2 =((π/4)+(1/2)(1))^2 =((π/4)+(1/2))^2

Commented byjohn_santu last updated on 08/Aug/21

false . lim_(x→0) ((π/4)+((sin x)/(2x)))^2 ≠ (π/4)+(1/2)