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Math Help - proving uniformly continousity

  1. #1
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    proving uniformly continousity

    i need to prove that  f(x)=ln(x^{2}+cos^{2}x) is uniformly continous. now, i thought of a way. i know that  f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x}
    so i acctually need to prove that  f'(x) is bounded (  N\leq f'(x)\leq M ). but how do i prove it?
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  2. #2
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    Re: proving uniformly continousity

    Quote Originally Posted by orir View Post
    i need to prove that  f(x)=ln(x^{2}+cos^{2}x) is uniformly continous. now, i thought of a way. i know that  f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x}
    so i acctually need to prove that  f'(x) is bounded (  N\leq f'(x)\leq M ). but how do i prove it?
    Have a look at the graph of the derivative.
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  3. #3
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    Re: proving uniformly continousity

    i took a look, and i can see that its limit in infinity (and -infinity) is 0 (is that what you're implying of?) . but as i posted in another post, i don't really know how to solve this limit by myself.
    is there any other way except the one with the limit?
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    Re: proving uniformly continousity

    Quote Originally Posted by orir View Post
    i took a look, and i can see that its limit in infinity (and -infinity) is 0 (is that what you're implying of?) . but as i posted in another post, i don't really know how to solve this limit by myself.
    is there any other way except the one with the limit?
    I was simply pointing out that the derivative is in fact bounded.

    That derivative is continuous. and the limit is 0 as x\to \pm\infty.
    Last edited by Plato; June 13th 2013 at 12:37 PM.
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  5. #5
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    Re: proving uniformly continousity

    but why is that so?
    and, is it enough to say it's bounded?
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  6. #6
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    Re: proving uniformly continousity

    Quote Originally Posted by orir View Post
    i need to prove that  f(x)=ln(x^{2}+cos^{2}x) is uniformly continous. now, i thought of a way. i know that  f'(x)=\frac{2x-sin2x}{x^{2}+cos^{2}x}
    so i acctually need to prove that  f'(x) is bounded (  N\leq f'(x)\leq M ). but how do i prove it?
    See the following very similar post:
    Uniform Continuity of $\sqrt{|x|}$ on $\mathbb{R}$
    and see "The more sophisticated way" part in
    Uniform Continuity of $\sqrt{|x|}$ on $\mathbb{R}$
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