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Math Help - Proof ArcTan limit

  1. #1
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    Proof ArcTan limit

    Hey,
    I wanted to ask how is it possible to proof that ArcTan(1/x^2) limit when x goes to infinity, is 0 ?
    I need a valid proof and not a intuitive proof (Epsilon Delta\ Or limit arithmetic ..)

    Thanks a lot
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  2. #2
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    Re: Proof ArcTan limit

    Well, arctangent is the inverse to tangent: y= arctan(1/x^2) is the same as 1/x^2= tan(y). I assume you know that as x goes to infinity, 1/x^2 goes to 0. So what must tan(y) be? And then, what must y be?

    Now that's "intuitive" and not a "valid proof" but it is thinking like that that will lead you to a "valid proof". We want to show that in order to make arctan(1/x^2) as close to 0 as we please, |arctan(1/x^2)|< \epsilon, which is -\epsilon< arctan(1/x^2)< \epsilon, 0 < 1/x^2< tan(\epsilon), x^2> tan(\epsilon), x> \sqrt{tan(\epsilon).

    So take N> \sqrt{tan(\epsilon)} and work backwards: if x> N then |arctan(1/x^2)|< \epsilon.
    Last edited by HallsofIvy; December 7th 2012 at 12:21 PM.
    Thanks from topsquark
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    Re: Proof ArcTan limit

    Quote Originally Posted by MichaelEngstler View Post
    Hey,
    I wanted to ask how is it possible to proof that ArcTan(1/x^2) limit when x goes to infinity, is 0 ?
    I need a valid proof and not a intuitive proof (Epsilon Delta\ Or limit arithmetic ..)

    You know that \arctan(x) is a continuous function.

    And \arctan(0)=0 also \lim _{x \to \infty } \frac{1}{x} = 0.

    Combine these \left( {\forall \varepsilon  > 0} \right)\left( {\exists \delta  > 0} \right)\left[ {\left| x \right| < \delta \to \left| {\arctan (x)} \right| < \varepsilon } \right].

    And \left( {\forall \delta  > 0} \right)\left( {\exists N \in \mathbb{N}} \right)\left[{x \geqslant N \to \frac{1}{x} < \delta } \right]
    Last edited by Plato; December 7th 2012 at 01:01 PM.
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    Re: Proof ArcTan limit

    Thanks both for your help,
    It was very helpful ! (:
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    Re: Proof ArcTan limit

    A question for HallsofIvy:
    On line 3, You got from 0<1/x^2<tan(e) to x^2>tan(e) .. shouldnt it be x^2 > 1/tan(e) ? And then the final requirement x>1/sqrt(tan(e)) ?
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  6. #6
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    Re: Proof ArcTan limit

    Yes, that was a typo. I intended to type cot(e).
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  7. #7
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    Re: Proof ArcTan limit

    I started to write the proof and found another problem, Say e = 2.5.
    I want N to be > 1/sqrt(tan(2.5)) .. But 1/sqrt(tan(2.5)) is undefined because tan(2.5) < 0.
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  8. #8
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    Re: Proof ArcTan limit

    Quote Originally Posted by MichaelEngstler View Post
    I started to write the proof and found another problem, Say e = 2.5.
    I want N to be > 1/sqrt(tan(2.5)) .. But 1/sqrt(tan(2.5)) is undefined because tan(2.5) < 0.
    If you work only with \arctan(t) you will avoid those kinds of problems.
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