Evaluate the limit if it exists. 1.lim x-->0 (sinx)^2/(sin(x^2)) 2.lim x-->0 (tan3x)/(tan pi x) 3.lim x-->0 (x-sinx)/(tanx-x)?

1.
as x → 0
sin x → x , so
(sin x)^2 / sin (x^2)
→ x^2 / x^2
→ 1 ..... Ans

2.
as x → 0
tan 3x / tan πx
→ 3*sec^2 3x / π*sec^2 πx , by the L'Hopital rule
→ 3/π ..... Ans

3.
as x → 0
( x - sin x ) / ( tan x - x )
→ ( 1 - cos x ) / ( sec^2 x - 1 ) , by the L rule
→ sin x / ( 2*sec x * sec x*tan x ) , L rule
= (1/2)sin x * cos^2 x * ( cos x / sin x )
= (1/2)cos^3 x
→ 1/2 ..... Ans