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daniel_i_l
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Homework Statement
let [tex] F(x) = {\frac{1}{x}}{\int^{x}_{0} e^{-t^2} dt} [/tex]
Prove, without calculating the integral (but assuming that F exists in (0,infinity)) that:
a) the limit of F at 0 is 1.
b) for all x>0 [tex] \int^{x}_{0} e^{-t^2} dt <= x[/tex]
x) supF((0,infinity)) = 1
Homework Equations
The Attempt at a Solution
a) I solved a using L'hopitals rule and the fundamental theorm.
b) First I defined a new function G where G(x) = F(x) for all x>0 and G(x) = 1 for x=0.
Now, I know that G is continues for all x>=0 and has a derivative for all x>0. So I have to prove that G'(x) < 0 for all x>0. I got that in order to prove that I have to prove that
[tex] \int^{x}_{0} e^{-t^2} dt > x e^{-x^2}[/tex] for all x>0. In order to do this I can define a new function [tex] H(x) \int^{x}_{0} e^{-t^2} dt - x e^{-x^2}[/tex]. Now, H(0)=0 and for all x>0 H'(x) >0 and so H(x)>0 for all x>0 which means that G'(x) < 0 for all x>0 and so G(x) < 1 for all x>0 and since in this domain G(x)=F(x) we get that F(x)<1 for all x>0.
But this is a stronger result than what was asked for (=<), is my proof right?
c)We know from b that 1 is an upper bound of F in (0,infinity). Now, let's assume that there exists a number 0<c<1 so that c is also an upper bound. Since the limit of F at 0 is 1 we can find a D (delta) so that for all 0<x<D |F(x)-1| < (1-c)/2. This means that
F(x)-1 > -(1-c)/2 => F(x) > c which contradicts the fact that c is an upper bound!
Are those right? Especially (b) and (c).
Thanks.
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