How do I solve this limit? lim_(x->0)((1-cos^3(3x))/(xsin6x))

I know I have to find a workaround the division by zero, but I just don't see it.

Thanks in advance.

1 Answer
Jim H
Mar 20, 2018

\ See below.

Explanation:

(1-cos^3(3x))/(xsin(6x)) = ((1-cos(3x)(1+cos(3x)+cos^2(3x))))/(2xsin(3x)cos(3x))

= ((1-cos(3x)(1+cos(3x)+cos^2(3x))))/(2xsin(3x)cos(3x)) * (1+cos(3x))/(1+cos(3x))

= (sin^2(3x)(1+cos(3x)+cos^2(3x)))/(2xsin(3x)cos(3x)(1+cos(3x))

= 3/2(sin(3x)/(3x))(1+cos(3x)+cos^2(3x))/(cos(3x)(1+cos(3x))

As xrarr0, we see the limit is 9/4

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