Question Number 101366 by bobhans last updated on 02/Jul/20

lim_(x→0)  ((x^3 −sin ^3 x)/x^5 ) =?

Commented byDwaipayan Shikari last updated on 02/Jul/20

lim_(x→0) ((x^3 −(x−(x^3 /6))^3 )/x^5 )=lim_(x→0) ((x^3 −x^3 +(x^9 /6)+((3x^5 )/6)−((3x^7 )/(36)))/x^5 )=(((3x^5 )/6)/x^5 )=(1/2)  {(x^9 /6) and ((3x^7 )/(36))are  negligable}

Commented bybemath last updated on 02/Jul/20

agree

Answered by ajfour last updated on 02/Jul/20

lim_(x→0) (((x−sin x)(x^2 +xsin x+sin ^2 x))/x^5 )  lim_(x→0) (((x−sin x))/x^3 )×lim_(x→0) (((x^2 +xsin x+sin ^2 x))/x^2 )  =lim_(x→0) ((x−(x−(x^3 /(3!))+....))/x^3 )×(3)  =  (1/(3!))×3 = (1/2) .

Commented bybemath last updated on 02/Jul/20

yes..thank you sir