How do you find the limit of (1-cos (4x))/(1-cos (3x)) as x approaches 0?

navelarp8c

navelarp8c

Answered question

2023-02-15

How to find the limit of 1 - cos ( 4 x ) 1 - cos ( 3 x ) as x approaches 0?

Answer & Explanation

Sdunkenf9cu

Sdunkenf9cu

Beginner2023-02-16Added 5 answers

The series expansion of cos ( α x ) for x = 0 in an open set containing 0 is given by i = 0 ( - 1 ) i ( α x ) 2 i 2 n !
Thus
1 - cos ( 4 x ) = 8 x 2 - 32 x 4 3 + 256 x 6 45 + ...
also
1 - cos ( 3 x ) = 9 x 2 2 - 27 x 4 8 + 81 x 6 80 + ...
Thus
1 - cos ( 4 x ) = x 2 ( 8 - 32 x 2 3 + 256 x 4 45 + ... )
also
1 - cos ( 3 x ) = x 2 ( 9 2 - 27 x 2 8 + 81 x 4 80 + ... )
Substituting in the fraction
1 - cos ( 4 x ) 1 - cos ( 3 x ) = ( 8 - 32 x 2 3 + 256 x 4 45 + ... ) ( 9 2 - 27 x 2 8 + 81 x 4 80 + ... )
computing the limit for x 0 we obtain
lim x 0 1 - cos ( 4 x ) 1 - cos ( 3 x ) = 16 9

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?