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Math Help - Limit wit compliacated (looking) intergrals

  1. #1
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    Limit wit compliacated (looking) intergrals

    Could you please help me solving this complicated (or maybe just complicated looking limit problem?

    \lim_{\displaystyle x \to 0+} \frac{\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\  sqrt{\displaystyle1+z^4}\mathrm{d}z\right) \mathrm{d}t}{\displaystyle\int_{0}^{x}\left(\int_{  0}^{t}\sqrt{\displaystyle1+z^6}\mathrm{d}z\right) \mathrm{d}t}

    Any help would be appreciated!
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  2. #2
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    You could try several approaches, I think. One is to use l"Hospital's rule. Another is to reverse the order of integration in both numerator and denominator. You should be able to get traction one way or the other. How does that strike you?
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  3. #3
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    Thank you for your help.
    I am afraid, I still can't solve the problem
    I uderstand that  \lim_{\displaystyle x \to 0+}\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\sqr  t{\displaystyle1+z^4}\mathrm{d}z\right) \mathrm{d}t=\lim_{\displaystyle x \to 0+}\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\sqr  t{\displaystyle1+z^6}\mathrm{d}z\right) \mathrm{d}t=0
    so we can use L'Hospital's rule. But I don't know how to derive these expressions (with respect to z?).
    So I need a little more help!
    Thank you!
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  4. #4
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    F(x) = \int_0^x {\int_0^t {\sqrt {1 + z^4 } dz} dt} \; \Rightarrow \;F'(x) = \int_0^x {\sqrt {1 + z^4 } dz} \; \Rightarrow \;F''(x) = \sqrt {1 + x^4 }
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  5. #5
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by doug View Post
    Could you please help me solving this complicated (or maybe just complicated looking limit problem?

    \lim_{\displaystyle x \to 0+} \frac{\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\  sqrt{\displaystyle1+z^4}\mathrm{d}z\right) \mathrm{d}t}{\displaystyle\int_{0}^{x}\left(\int_{  0}^{t}\sqrt{\displaystyle1+z^6}\mathrm{d}z\right) \mathrm{d}t} Any help would be appreciated!
    Applying two times de l'Hopital's rule You obtain...

    \displaystyle \lim_{x \rightarrow 0+}  \frac{\int_{0}^{x} \int_{0}^{t} \sqrt{1+z^{4}}\ dz\ dt}{\int_{0}^{x} \int_{0}^{t} \sqrt{1+z^{6}}\ dz\ dt} = \lim_{x \rightarrow 0+}  \frac{\int_{0}^{x} \sqrt{1+z^{4}}\ dz} {\int_{0}^{x} \sqrt{1+z^{6}}\ dz}=

    \displaystyle = \lim_{x \rightarrow 0+} \frac{\sqrt{1+x^{4}}}{\sqrt{1+x^{6}}} = 1

    Kind regards

    \chi \sigma
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