Find the limit as x approaches 0. 1-cos3x/sin3x.

The main formula is lim (1 - cos X) / X = 0, when X approaches 0, and

lim sinX / X = 1 when X approaches 0

lim (1-cos3x/sin3x) = lim (3x/3x) (1-cos3x/sin3x) = lim [ (1-cos3x) / 3x].[lim

X approaches 0 X approaches 0 X approaches 0

1 / (sin3x/3x) ] = 0 x 1/1=0

finally lim (1-cos3x/sin3x) = 0

X approaches 0

The limit as x approches 0 of (1 - cos3x) / sin3x = the limit as x approches 0 of 3sin 3x / 3cos 3x = 3sin 0 / 3 cos 0 = 0/3 = 0.