Re: lim ( f(x)/x )


"stat" <stat@xxxxxxx> wrote in message
news:h%__e.92720$qY1.36454@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Please so me if there are any mistakes in the "proof" that follows and
> also how to extend it by not assuming f nondecreasing
> Thanks
>
> Given :
> 1. f (x) differentiable and nondecreasing over R
> 2. lim f ' (x) = 0 as x->infinity
>
> Show :
> lim ( f (x) / x ) -> 0 as x->infinity
>

For an easier proof, think about L'Hopital's rule.

> "Proof" :
> 2 => ( f(x) - f(xo) ) / (x - xo) < e if x > xo >N and x-xo < delta
>
> after multiplying both sides by (x-xo) we get
>
> (f(x) - f(xo) )< e * (x - xo)
>
> after adding f(xo) to both sides we get
>
> f(x) < e*(x-xo) + f (xo) = e*x - e*xo + f(xo)
>
> after dividing both sides by x we get
>
> 0 < f(x)/x < e - e*xo/x + f(xo)/x
>
> e can be made as small as we please and xo and f(xo) are finite real
> numbers therefore
> as x->infinity the RHS converges to 0 so f(x)/x converges to 0
>
>
>
>


.

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