Question

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

Answer 1

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

Answer 2

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.