If f(x)=1/3[f(x+1) + {5/f(x+2)}]
Then evaluate lim f(x) where x tends to infinity.
(F(x)0 for all x€R.

Question

If f(x)=1/3[f(x+1) + {5/f(x+2)}] 
Then evaluate lim f(x) where x tends to infinity.
(F(x)>0  for all x€R.

samigupta8 · Answer

@ganeshie8 sir r u there?

ganeshie8 · Answer

Assume that the limit exists

ganeshie8 · Answer

Let $\lim\limits_{x\to\infty}f(x) = L$

ganeshie8 · Answer

Would you agree that
Let $\lim\limits_{x\to\infty}f(x) = \lim\limits_{x\to\infty}f(x+1) = \lim\limits_{x\to\infty}f(x+5) =   L$

?

ganeshie8 · Answer

Since the sequence $f(n)$ converges, the value of $f(n)$ will be close to $L$ for all large values of $n$ and in the limiting case, they all are equal to $L$

ganeshie8 · Answer

$$f(x)=1/3[f(x+1) + {5/f(x+2)}] $$

take the limit both sides and get

$$L=1/3[L+ 5/L] $$

you can solve $L$

samigupta8 · Answer

Wait please!
I m not getting how u came up with limits of f(x) to b equal to that for f(x+1) and f(x+2)...

ganeshie8 · Answer

as an example, maybe consider
$$f(x) = \dfrac{3x+2}{2x+1}$$

$f(x+1) = ?$
$f(x+2) = ?$

$\lim\limits_{x\to\infty}f(x)=?$
$\lim\limits_{x\to\infty}f(x+1)=?$
$\lim\limits_{x\to\infty}f(x+2)=?$

samigupta8 · Answer

Did u literally made up for a function that satisfies the condition of f(x) to be true?
(I mean it might have took a great deal of tym) :)

samigupta8 · Answer

Btw.
Is there a theoretical way also to prove that limf(x) =Limf(x+1)?

ganeshie8 · Answer

I did not make up that function. 
Any function that has a finite limit satisfies the above relation

ganeshie8 · Answer

the proof is not hard
but do you get the intuition on why f(x) and f(x+a) will have the same limiting value ?

samigupta8 · Answer

Maybe bcoz if we substitute x as infinity in the f(x) we get f(infinity) and similarly x+1 would be infinity.
So f(x+1) will be also f(infinity) so we can say that the two r identical.

ganeshie8 · Answer

that may do for now

in order to get more sense of this, you will need to prove it rigorously using epsilon-delta definition or sequences definition of the limit

ganeshie8 · Answer

i suggest you not get into them now...

samigupta8 · Answer

Okay! 
This will work for now.(what i told)
Ryt?

ganeshie8 · Answer

Yes, as x -> infinity, we can say that x + 1 is same as x + 2

samigupta8 · Answer

Ty.

ganeshie8 · Answer

yw