Trig Limit Problems? Ten Points to the first best answer?

Question

Limit a -> 0 (sin(3a))/(sin(2a))

Limit a -> 0 (cos((pi/2) + a))/a

Limit a -> 0 (a^2)/(1 - cos^2(a))

I've been stuck for two HOURS!!!!!!!!!!!!!!!!!!!!

any help is really really appreciated!!!!!!!!

Answer 1

lim a → 0 sin3a / sin2a

= lim a → 0 [(sin3a / 3a) / (sin2a/ 2a)] * (3/2)

= 3/2.

lim a → 0 cos(π/2+ a) / a

= lim a → 0 - sina / a

= -1.

lim a → 0 a² / (1 - cos² a)

= lim a → 0 a² / sin² a

= 1.

Answer 2

lim [ sin(3a) / sin(2a) ]

a-->0

= lim [ (sin3a / a) * (a / sin2a) ]

= lim [sin(3a) / a] * lim [a / sin2a]

= lim [ 3*sin(3a)/3a] * lim[ 1/2 (2a/sin2a) ]

= 3 * 1/2     [ because lim (sinx / x) = 1]

                                 x-->0

=3/2.

2) lim [cos (pi/2 + a) / a]

    a-->0

= lim [- sina / a]

= -1.

3) lim [ a^2 / (1-cos^2a)]

    a-->0

= lim [a^2 / sin^2a]

= [ lim (a / sina)]^2

= 1^2

= 1.