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badreferences

Bored?\[f:\mathbb R\to\mathbb R\mid e^{x-\int f\left(x\right)}=\int f\left(x\right)\]Find:\[\lim_{x\to\infty}\left(f\left(x\right)\right)^x\]

  • 2 years ago
  • 2 years ago

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  1. LifeIsADangerousGame
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    The answer is 0. And I'm still bored -_-

    • 2 years ago
  2. LifeIsADangerousGame
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    Haha! I'm right aren't I?

    • 2 years ago
  3. badreferences
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    No, lol, I hope that isn't a real question. XD

    • 2 years ago
  4. LifeIsADangerousGame
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    -_- You are clearly misunderstanding me <-- (see what I did there?) The answer is F.

    • 2 years ago
  5. satellite73
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    i don't even understand the question

    • 2 years ago
  6. badreferences
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    Also, I should've specified a \(dx\) in the integrals, so it's clear they aren't an operation of \(f\). But I think you all knew that already.

    • 2 years ago
  7. LifeIsADangerousGame
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    I didn't. So you misunderstood me again :D

    • 2 years ago
  8. FoolForMath
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    what is \( dx \)? :P

    • 2 years ago
  9. badreferences
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    We're given conditions \(f:\mathbb R\to\mathbb R\) which means the function maps reals into reals. We know that \(\int f\left(x\right)\,dx=e^{x-\int f\left(x\right)\,dx}\). We want to find the limit asked knowing this much.

    • 2 years ago
  10. LifeIsADangerousGame
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    Could you repeat that?

    • 2 years ago
  11. badreferences
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    @LifeIsADangerousGame You are a snarky one, aren't you?

    • 2 years ago
  12. LifeIsADangerousGame
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    It's my job. It's what I do.

    • 2 years ago
  13. Ishaan94
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    Solution? ಠ_ಠ

    • 2 years ago
  14. Ishaan94
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    @Mr.Math SAVVEE MEE!

    • 2 years ago
  15. Mr.Math
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    @Ishaan94 I will come back to it later and see if I can do it. I feel tired now.

    • 2 years ago
  16. Ishaan94
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    \[\mathsf{ \color{yellowgreen}{\text{Okay}}}\]

    • 2 years ago
  17. Mr.Math
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    By taking ln of both sides we have \[x-\int f(x)dx=\ln(\int f(x)dx)) \implies 1-f(x)=\frac{f(x)}{\int f(x)dx}\] \[\large \implies f(x)=\frac{\int f(x)dx}{\int f(x)dx+1}.\] Because \(\int f(x)dx>0\), we have \( 0<f(x)<1\). Hence \[\large \lim_{x\to \infty} (f(x))^x=0.\]

    • 2 years ago
  18. Ishaan94
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    Oh, I did the same but read above, badref rejected Zero as the answer :/

    • 2 years ago
  19. karatechopper
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    ISHAAN GET IN CHAT IF U R NOT HELPIN...

    • 2 years ago
  20. karatechopper
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    :D intrestin lookin R u got there

    • 2 years ago
  21. Mr.Math
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    Well, he/she should post the answer then.

    • 2 years ago
  22. Ishaan94
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    Post the Answer! Post the Answer! Post the Answer!

    • 2 years ago
  23. badreferences
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    Lol, I actually mistyped the question. I think the answer for this is actually \(0\), my bad.

    • 2 years ago
  24. Mr.Math
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    What is the question then?

    • 2 years ago
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