Question Number 149251 by john_santu last updated on 04/Aug/21

lim_(x→0^+ ) ((tan x (√(tan x))−sin x (√(sin x)))/(x^3 (√x))) =?

Answered by dumitrel last updated on 04/Aug/21

lim_(x→0^+ ) ((sinx)/x)∙lim_(x→0^0 ) (√((sinx)/x))∙lim_(x→0^+ ) (1/x^2 )((1/(cosx(√(cosx))))−1)= lim_(x→0^+ ) ((1−cos^3 x)/(x^2 cosx(√(cosx))(1+cosx(√(cosx)))))=lim_(x→0^+ ) (((1−cosx)(1+cosx+cos^2 x))/(x^2 cosx(√(cosx))(1+cosx(√(cosx)))))= =lim_(x→0^+ ) ((2sin^2 (x/2)(1+cosx+cos^2 x))/(x^2 cosx(√(cosx))(1+cosx(√(cosx)))))= =lim_(x→0^+ ) (((sin(x/2))/(x/2)))^2 (((1+cosx+cos^2 x))/(2cosx(√(cosx))(1+cosx(√(cosx)))))=(3/4)

Answered by john_santu last updated on 04/Aug/21