lim as x approaches 0

Question

(1-cos(2x))/(3sinx)

How do I derive both the numerator and denominator

Answer 1

using L'Hopital's rule

lim (1-cos(2x))/(3sinx) as x ---> 0
= lim 2sin2x/3cosx
= 4sinxcosx/3cosx
= (4/3)sinx , as x --->
= 0

Answer 2

How did you get 4sinxcosx

Answer 3

sin(2A) = 2sinA cosA, one of the standard identities.

Answer 4

So that's saying sin(2x)=2sinxcosx?

What if there was no 2 in sin(2x)? I dont rememeber the identities is there a link to all the identities somewhere

Answer 5

have you seen
sin(A+B) = sinAcosB + cosAsinB ???
then sin (2A) = sin(A+A)
= sinAcosA + cosAsinA
= 2sinAcosA, called the double angle formula.

Answer 6

No we haven't went over that identity in my class. So I was wondering if there was another way to solve this question

Answer 7

could just go from:
lim 2sin2x/3cosx as x----> 0
= 2(0)/(3(1))
= 0/3
= 0