Here's the question you clicked on:
badreferences
Bored?\[f:\mathbb R\to\mathbb R\mid e^{x-\int f\left(x\right)}=\int f\left(x\right)\]Find:\[\lim_{x\to\infty}\left(f\left(x\right)\right)^x\]
The answer is 0. And I'm still bored -_-
Haha! I'm right aren't I?
No, lol, I hope that isn't a real question. XD
-_- You are clearly misunderstanding me <-- (see what I did there?) The answer is F.
i don't even understand the question
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.
I didn't. So you misunderstood me again :D
what is \( dx \)? :P
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.
Could you repeat that?
@LifeIsADangerousGame You are a snarky one, aren't you?
It's my job. It's what I do.
@Ishaan94 I will come back to it later and see if I can do it. I feel tired now.
\[\mathsf{ \color{yellowgreen}{\text{Okay}}}\]
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 \[\large \lim_{x\to \infty} (f(x))^x=0.\]
Oh, I did the same but read above, badref rejected Zero as the answer :/
ISHAAN GET IN CHAT IF U R NOT HELPIN...
:D intrestin lookin R u got there
Well, he/she should post the answer then.
Post the Answer! Post the Answer! Post the Answer!
Lol, I actually mistyped the question. I think the answer for this is actually \(0\), my bad.
What is the question then?