Limit of in equation

[SolidSnake]

Hello everyone

I have a wired Problem and my school maths cannot handle it.

There are two expressions:


(1-a)X

and

[X/(1+a)^2] + [e(2+a)/(1+a)]

X can be set to any Value greater than 0.
a is defined by an open Intervall (0 ; 1)
e is an infinitesimally small positive number

I want to know for which values of a the inequation

(1-a)X > [X/(1+a)^2] + [e(2+a)/(1+a)]

holds true.

I thought of taking the limit of both sides with e -> 0 so that

(1-a)X > [X/(1+a)^2]

this can be easily solved.
But is it legal to take the limit of an in equation?

 

[HallsofIvy]

I thought it was peculiar to have two consecutive prepositions, "of" and "in"! You mean "inequation". (I would say "inequality".) I don't see any reason to take a limit. The question asks about the inequality as it stands, not with a limit. You have (1- a)X> X/(1+ a)^2+ e(2+ a)/(1+ a). I would clear the fractions by multiply both sides by the positive (1+ a)^2. That gives (1- a)(1+ a^2)X> X+ e(2+ a)(1+ a). Subtract X+ e(2+ a)(1+a) from both sides to get (1- a)(1+ a^2)X- e(2+ a)(1+ a)- X> 0. The left side is a cubic in a. There is not going to be any reasonable way to solve that but, since e is small, you can approximate the solution by taking e= 0 (I would not say "take the limit" but that is essentially the same) to get (1- a)(1+ a^2)- 1> 0.

 

[SolidSnake]

@HallsofIvy thank you fore your response. I am very sorry form my terminology. I am not a native have no higher education in maths. I just try it for fun.

So you would say as a proof it is sufficient zu solve the cubic and approximate the e term with zero.
I thought of taking the limit because e should be es small es possible but still positive...