Solve this limit. limx rarr 0 ((e^xsinx−(x+1)tanx))/((xlogcos(x)))

cottencintu

cottencintu

Answered question

2022-08-10

Solve this limit.
lim x 0 ( e x sin x ( x + 1 ) tan x ) ( x log cos ( x ) )

Answer & Explanation

kidoceanoe

kidoceanoe

Beginner2022-08-11Added 15 answers

Expanding to the 3rd order
x log cos ( x ) = x 3 2 + o ( x 3 )
e x sin x = ( 1 + x + x 2 / 2 + o ( x 2 ) ) ( x x 3 / 6 + o ( x 3 ) ) = x + x 2 + x 3 / 3 + o ( x 3 )
( x + 1 ) tan x = ( x + 1 ) ( x + x 3 / 3 + o ( x 3 ) ) = x + x 2 + x 3 / 3 + o ( x 3 )
thus
lim x 0 ( e x sin x ( x + 1 ) tan x ) ( x log cos ( x ) ) = lim x 0 o ( x 3 ) x 3 2 + o ( x 3 ) = lim x 0 o ( 1 ) ) 1 2 + o ( 1 ) = 0

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