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

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LifeIsADangerousGame
 2 years ago
Best ResponseYou've already chosen the best response.0The answer is 0. And I'm still bored _

LifeIsADangerousGame
 2 years ago
Best ResponseYou've already chosen the best response.0Haha! I'm right aren't I?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1No, lol, I hope that isn't a real question. XD

LifeIsADangerousGame
 2 years ago
Best ResponseYou've already chosen the best response.0_ You are clearly misunderstanding me < (see what I did there?) The answer is F.

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0i don't even understand the question

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Also, 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
 2 years ago
Best ResponseYou've already chosen the best response.0I didn't. So you misunderstood me again :D

FoolForMath
 2 years ago
Best ResponseYou've already chosen the best response.0what is \( dx \)? :P

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1We'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
 2 years ago
Best ResponseYou've already chosen the best response.0Could you repeat that?

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1@LifeIsADangerousGame You are a snarky one, aren't you?

LifeIsADangerousGame
 2 years ago
Best ResponseYou've already chosen the best response.0It's my job. It's what I do.

Mr.Math
 2 years ago
Best ResponseYou've already chosen the best response.1@Ishaan94 I will come back to it later and see if I can do it. I feel tired now.

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.3\[\mathsf{ \color{yellowgreen}{\text{Okay}}}\]

Mr.Math
 2 years ago
Best ResponseYou've already chosen the best response.1By taking ln of both sides we have \[x\int f(x)dx=\ln(\int f(x)dx)) \implies 1f(x)=\frac{f(x)}{\int f(x)dx}\] \[\large \implies f(x)=\frac{\int f(x)dx}{\int f(x)dx+1}.\] Because \(\int f(x)dx>0\), we have \( 0<f(x)<1\). Hence \[\large \lim_{x\to \infty} (f(x))^x=0.\]

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.3Oh, I did the same but read above, badref rejected Zero as the answer :/

karatechopper
 2 years ago
Best ResponseYou've already chosen the best response.0ISHAAN GET IN CHAT IF U R NOT HELPIN...

karatechopper
 2 years ago
Best ResponseYou've already chosen the best response.0:D intrestin lookin R u got there

Mr.Math
 2 years ago
Best ResponseYou've already chosen the best response.1Well, he/she should post the answer then.

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.3Post the Answer! Post the Answer! Post the Answer!

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Lol, I actually mistyped the question. I think the answer for this is actually \(0\), my bad.

Mr.Math
 2 years ago
Best ResponseYou've already chosen the best response.1What is the question then?
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