Limit evaluate lim_(x->0)((ln( cos(-5x)))/( ln( cos(-3x))))

paratusojitos0yx

paratusojitos0yx

Answered question

2022-11-17

Limit evaluate lim x 0 ln ( cos ( 5 x ) ) ln ( cos ( 3 x ) )
Please help me with this limit without using L'Hôpital's rule. I would by happy if you use simple solving. Thank you as much as I can ;).

Answer & Explanation

Faith Wise

Faith Wise

Beginner2022-11-18Added 17 answers

We have the following Taylor series approximations about zero ( v = 0 and w = 0):
ln ( 1 + w ) = w + O ( w 2 ) w
and
cos ( v ) = 1 1 2 v 2 + O ( v 4 ) 1 1 2 v 2
Numerator: Put v = 5 x to get cos ( 5 x ) 1 25 2 x 2
and then put w = 25 2 x 2 to get ln ( cos ( 5 x ) ) 25 2 x 2
Denominator: Put v = 3 x to get cos ( 3 x ) 1 9 2 x 2 and then put w = 9 2 x 2 to get ln ( cos ( 3 x ) ) 9 2 x 2
Hence, the limit is
L = lim x 0 25 2 x 2 9 2 x 2 = 25 9

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