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Math Help - how to prove that this function is bounded..

  1. #1
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    how to prove that this function is bounded..

    f(x)=(1+x+x^2 +x^3 +x^4)/(1+x^4)
    using lmit n->+infinity
    obviously this function goes to 1

    is that all i need to do in order to show that its bounded
    ??
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by transgalactic View Post
    f(x)=(1+x+x^2 +x^3 +x^4)/(1+x^4)
    using lmit n->+infinity
    obviously this function goes to 1

    is that all i need to do in order to show that its bounded
    ??
    Case #1 1\leqslant x

    \begin{aligned}\frac{1+x+x^2+x^3+x^4}{1+x^4}&\leqs  lant\frac{x^4+x^4+x^4+x^4+x^4}{1+x^4}\\<br />
&\leqslant\frac{5x^4}{x^4}\\<br />
&=5\end{aligned}

    Case #2 0\leqslant x <1

    \begin{aligned}\frac{1+x+x^2+x^3+x^4}{1+x^4}&\leqs  lant\frac{1+x+x+x+x}{1+x^4}\\<br />
&\leqslant\frac{4+4x}{1+x^4}\\<br />
&\leqslant\frac{4(1+x)}{1-x^4}\\<br />
&=\frac{4}{1-x+x^2-x^3}\\<br />
&\leqslant 4\end{aligned}

    Is that what you wanted?

    EDIT: Using calculus you can greatly improve this to f(x)<.7
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by transgalactic View Post
    f(x)=(1+x+x^2 +x^3 +x^4)/(1+x^4)
    using lmit n->+infinity
    obviously this function goes to 1

    is that all i need to do in order to show that its bounded
    ??
    As the denominator has no zeros \mathbb{R} and the numerator is a polynomial, it is sufficient to show that the two limits as x \to \pm \infty are both finite.

    CB
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  4. #4
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    solved it thanks
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