anonymous
  • anonymous
Bored?\[f:\mathbb R\to\mathbb R\mid e^{x-\int f\left(x\right)}=\int f\left(x\right)\]Find:\[\lim_{x\to\infty}\left(f\left(x\right)\right)^x\]
Mathematics
schrodinger
  • schrodinger
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LifeIsADangerousGame
  • LifeIsADangerousGame
The answer is 0. And I'm still bored -_-
LifeIsADangerousGame
  • LifeIsADangerousGame
Haha! I'm right aren't I?
anonymous
  • anonymous
No, lol, I hope that isn't a real question. XD

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LifeIsADangerousGame
  • LifeIsADangerousGame
-_- You are clearly misunderstanding me <-- (see what I did there?) The answer is F.
anonymous
  • anonymous
i don't even understand the question
anonymous
  • anonymous
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.
LifeIsADangerousGame
  • LifeIsADangerousGame
I didn't. So you misunderstood me again :D
anonymous
  • anonymous
what is \( dx \)? :P
anonymous
  • anonymous
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.
LifeIsADangerousGame
  • LifeIsADangerousGame
Could you repeat that?
anonymous
  • anonymous
@LifeIsADangerousGame You are a snarky one, aren't you?
LifeIsADangerousGame
  • LifeIsADangerousGame
It's my job. It's what I do.
anonymous
  • anonymous
Solution? ಠ_ಠ
anonymous
  • anonymous
@Mr.Math SAVVEE MEE!
Mr.Math
  • Mr.Math
@Ishaan94 I will come back to it later and see if I can do it. I feel tired now.
anonymous
  • anonymous
\[\mathsf{ \color{yellowgreen}{\text{Okay}}}\]
Mr.Math
  • Mr.Math
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
anonymous
  • anonymous
Oh, I did the same but read above, badref rejected Zero as the answer :/
karatechopper
  • karatechopper
ISHAAN GET IN CHAT IF U R NOT HELPIN...
karatechopper
  • karatechopper
:D intrestin lookin R u got there
Mr.Math
  • Mr.Math
Well, he/she should post the answer then.
anonymous
  • anonymous
Post the Answer! Post the Answer! Post the Answer!
anonymous
  • anonymous
Lol, I actually mistyped the question. I think the answer for this is actually \(0\), my bad.
Mr.Math
  • Mr.Math
What is the question then?

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