Question

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

Answer 1

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

Answer 2

Haha! I'm right aren't I?

Answer 3

No, lol, I hope that isn't a real question. XD

Answer 4

-_- You are clearly misunderstanding me <-- (see what I did there?) The answer is F.

Answer 5

i don't even understand the question

Answer 6

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.

Answer 7

I didn't. So you misunderstood me again :D

Answer 8

what is \( dx \)? :P

Answer 9

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.

Answer 10

Could you repeat that?

Answer 11

@LifeIsADangerousGame You are a snarky one, aren't you?

Answer 12

It's my job. It's what I do.

Answer 13

Solution? ಠ_ಠ

Answer 14

@Mr.Math SAVVEE MEE!

Answer 15

@Ishaan94 I will come back to it later and see if I can do it. I feel tired now.

Answer 16

\[\mathsf{ \color{yellowgreen}{\text{Okay}}}\]

Answer 17

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
\[\large \lim_{x\to \infty} (f(x))^x=0.\]

Answer 18

Oh, I did the same but read above, badref rejected Zero as the answer :/

Answer 19

ISHAAN GET IN CHAT IF U R NOT HELPIN...

Answer 20

:D intrestin lookin R u got there

Answer 21

Well, he/she should post the answer then.

Answer 22

Post the Answer! Post the Answer! Post the Answer!

Answer 23

Lol, I actually mistyped the question. I think the answer for this is actually \(0\), my bad.

Answer 24

What is the question then?