# How to compute the limit of an infinitely nested function in Wolfram Alpha?

Take the function:

f(x) = sqrt( ... sqrt(sqrt(sqrt(sqrt(sqrt(x)-1)+1)-1)+1)-1 ...) +1

I want to find the limit of the function for a given value of x as the number of iterations tends towards infinity.

How? I am using Wolfram Alpha, could not get the "Nest" functions to work (and unsure if they'd help anyway).

• Can you use mathematica? Bacause questions about WA syntax are off topic. – Kuba Jul 7 '17 at 6:07

You can use Nest as follows:

Nest[Sqrt[Sqrt[#] - 1] + 1&, 2., 100]


1.4534

Or, you can use the usual trick for these kinds of things:

NSolve[r == Sqrt[Sqrt[r] - 1] + 1, r]


{{r -> 1.4534}, {r -> 1.}}

• Thanks. Nest function sadly doesn't work on WA but I thought it was a general Wolfram stack exchange. Your answer is correct for Mathematica so I'll check it. (I ended up just doing it manually on my TI 89 lol). – CommaToast Jul 7 '17 at 7:12
• @CommaToast The following WA input works for me. – Carl Woll Jul 7 '17 at 18:17

There are 2 fixed points. You can solve and illustrate,e.g.:

Solve[f[x] == x, x]
f[x_] := Sqrt[x - 1] + 1
nf[x_, n_] := Nest[f, x, n]
Manipulate[
DiscretePlot[nf[x0, j], {j, 1, 10, 1}, GridLines -> {None, {2}},
PlotRange -> {0, 5}], {x0, 1.1, 20, Appearance -> "Labeled"}]