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boloman0z

boloman0z

Answered question

2022-06-26

Find lim x 0 1 x ( 1 sin ( x ) 1 sinh ( x ) )

Answer & Explanation

trajeronls

trajeronls

Beginner2022-06-27Added 21 answers

HINT:
Instead of using L'Hospital's Rule, we can apply Taylor's Theorem and obtain
1 x ( 1 sin ( x ) 1 sinh ( x ) ) = 1 x ( sinh ( x ) sin ( x ) sin ( x ) sinh ( x ) ) = ( x + 1 6 x 3 + O ( x 5 ) ) ( x 1 6 x 3 + O ( x 5 ) ) x 3 + O ( x 5 )
Gaaljh

Gaaljh

Beginner2022-06-28Added 7 answers

A little bit of algebra:
1 x ( 1 sin ( x ) 1 sinh ( x ) ) = 1 x ( sinh ( x ) sin ( x ) sin ( x ) sinh ( x ) ) = x sin x x sinh x sinh x x + x sin x x 3 = x sin x x sinh x ( sinh x x x 3 + x sin x x 3 )
All the factors have a finite limit, so you can substitute each with its limit and find the answer you need.

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