Prove or disprove, limits of antiderivatives If g(x), f(x) are real and continuous, and g (

Dennis Montoya

Dennis Montoya

Answered question

2022-05-09

Prove or disprove, limits of antiderivatives
If g(x), f(x) are real and continuous, and g ( x ) f ( x ) g ( x ) + 1 x for every x > 1. The following limit is equal to 0:
lim x x x + x ( f ( t ) g ( t ) ) d t = 0

Answer & Explanation

Skyler Barber

Skyler Barber

Beginner2022-05-10Added 17 answers

Step 1
If: g ( x ) f ( x ) g ( x ) + 1 x
then: 0 f ( t ) g ( t ) 1 t
so: x x + x 0 d t x x + x f ( t ) g ( t ) d t x x + x d t t .
Step 2
Now integrate all parts and you get:
0 x x + x f ( t ) g ( t ) d t ln ( x + x ) ln ( x ) = ln ( 1 + 1 x ) now take the limit...
Daphne Haney

Daphne Haney

Beginner2022-05-11Added 4 answers

Explanation:
0 f ( t ) g ( t ) 1 t for t > 1, so if you integrate....

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