True or false? If true, justify. If false, give counter-example. If f,g:R→R are functions such that f is bounded and positive and lim_(x->+oo) g(x)=+oo, so lim_(x->+oo) f(x)g(x)=+oo

Bairaxx

Bairaxx

Answered question

2022-10-25

True or false? If true, justify. If false, give counter-example. If f , g : R R are functions such that f is bounded and positive and lim x + g ( x ) = + , so lim x + f ( x ) g ( x ) = +

Answer & Explanation

Kason Gonzales

Kason Gonzales

Beginner2022-10-26Added 15 answers

Let g ( x ) be anything where where lim x 0 g ( x ) = (for example g ( x ) = x). Let f ( x ) be any function so that f ( x ) is generally 1 g ( x ) . Okay we mus make some conditions so that f is always bounded and positive, so for example
f ( x ) = { 1 g ( x ) < 1 1 g ( x ) g ( x ) 1
The f ( x ) is bounded 0 < f ( x ) 1
f ( x ) g ( x ) = { g ( x ) g ( x ) < 1 1 g ( x ) 1
And lim x f ( x ) g ( x )
lim x f ( x ) g ( x ) = lim x 1 g ( x ) g ( x ) = 1 (Because there is an M so that g ( x ) > 1 for all x > M).
Marley Meyers

Marley Meyers

Beginner2022-10-27Added 3 answers

Counter example: f ( x ) = e x 2 , g ( x ) = x 2 .

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