limit trigonometry

Question

find the limit x => o from (tan x - x)/(x - sin x)

Answer 1

lim(x-->0)(tanx-x)/(x-sinx)

this is in 0/0 form

apply L hospital rule

= lim(x--->0)d/dx(tanx-x)/d/dx(x-sinx)

= lim(x--->0)(sec^2x-1)/(1-cosx)

Again apply L hospital rule.

= lm(x-->0)(2sec^2xtanx)/(sinx)

= lim(x-->0)[2sec^2x*sinx/cosx]/(sinx)

= lim(x---.0)[2sec^2x*1/cosx]

= lim (x--->0)[2sec^2x*secx]

= lim(x--->0)[2sec^3x]

By evaluting x = 0

= 2