Question

Question #553b2

Answer 1

`lim_(x->0)(1-cosx)/sin^2(3x) =1/18`

Explanation:

FIrst you can note that:

`(1-cosx)/sin^2(3x)= (1-cosx)/(3x)^2*((3x)/sin(3x))^2`

so that:

`lim_(x->0)(1-cosx)/sin^2(3x)=1/9*lim_(x->0)(1-cosx)/x^2*lim_(x->0)((3x)/sin(3x))^2`

But we know that:

`lim_(x->0)((3x)/sin(3x))^2 =1`

and :

`lim_(x->0)(1-cosx)/x^2=1/2`

Finally:

`lim_(x->0)(1-cosx)/sin^2(3x) =1/18`