- #1
terryds
- 392
- 13
Homework Statement
Find the limit of :
lim x-> (π/2) (2-2sin x)/(6x-3π)
2. The attempt at a solution
lim x-> (π/2) (2-2sin x)/(6x-3π)
=lim x-> (π/2) 2-2 sin x / 6 (x- (1/2)pi)
Assuming that y = x - (π/2)
So,
lim y->0 (2-2sin(y+pi/2))/6y
lim y->0 (2-2 (sin y cos pi/2 + cos y sin pi/2)/6y
Then, I substitute y=0 into the sin y cos pi/2 , so the equation remains :
lim y-> 0 (2-2 cos y sin pi/2)/6y
lim y->0 2(1-cos y sin pi/2) /6y
lim y-> 0 2 (2sin^2((1/2)y) sin pi/2) /6y
lim y->0 (4 sin^2 ((1/2)y) sin(pi/2)) /6y
lim y->0 (2/3) (sin^2 (1/2)y) (2*(1/2)y) (sin(pi/2))
Then, I'm stuck here, because I need one more (1/2)y as the denominator of sin(1/2)y , because sin^2(1/2)y = sin(1/2)y * sin(1/2)y, so I need two (1/2) y as the denominator
But, I can just make one. If I make two and normalize it, it'll be zero.. But, I'm not quite sure..
Please help me..