Let a , b , c , d be four distinct roots of x 4 − 4 x + 3 = x ( x 3 − f ′ ( 1 ) x 2 + f ′′ ( 1 ) x − 4 ) + f ( 1 ) , where f is a monic cubic. Comparing the identity gives f ′ ( 1 ) = f ′′ ( 1 ) =

Question

Let a , b , c , d be four distinct roots of x 4 − 4 x + 3 = x ( x 3 − f ′ ( 1 ) x 2 + f ′′ ( 1 ) x − 4 ) + f ( 1 ) , where f is a monic cubic. Comparing the identity gives f ′ ​ ( 1 ) = f ′′ ​ ( 1 ) = 0 and f ( 1 ) = 3 , hence f ( x ) = ( x − 1 ) 3 + 3 . Find lim x → 1 ​ ( f ( x ) − 3 ) 5/3 ( x − 1 ) 3 − s i n 3 ( x − 1 ) ​ .

TestHub · Accepted Answer

Put u = x − 1 . The denominator is ( u 3 ) 5/3 = u 5 . Since sin u = u − u 3 /6 + O ( u 5 ) , sin 3 u = u 3 − u 5 /2 + O ( u 7 ) . The quotient tends to 1/2.

Mathematics - Definite Integration Question with Solution | TestHub

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