Easy Math Question to most, but i need help!?

Question

Evaluate the limit of the trigonometric function:

lim (as X approaches Zero) 1-Cos3X / 7XsinX = ?????

Answer 1

Are you familiar with l'Hopital's Rule?

Is this at the calculus level?

both top and bottom -> 0 as x -> 0 so you can use:

limit (F/G) = limit (F '/ G ')

F = 1 - cos(3x)  ..... F ' = 3sin(3x)

G = 7xsin(x) ......... G ' = 7[sin(x) + xcos(x)]

Again both of these functions -> 0 as x -> 0

So you can apply the rule once again and use the second derivative.

F ' ' = 9cos(3x)

G ' ' = 7[cos(x) + cos(x) + xsin(x)] = 7[2cos(x) + xsin(x)]

Neither of these is zero so the limit is just:

9cos(3x)/{7[2cos(x) + xsin(x)]} evaluated as x -> 0

9/14

The limit is 9/14 = 0.642857