Finding limit of infinite term

[phasair]

Homework Statement

The problem is not homework, just something that has been bothering me. It's in picture form, in the attachment.
I'm an undergraduate in math, and this type of convergence is a bit unknown to me. What I've seen so far are normal sequences and series, and power series.
This however, is different, because there is no 'first' term, no 'beginning', so I don't really know how to approach this.

Homework Equations

The Attempt at a Solution

The equation I've found that describes this term is:

f(x)^2 = 1 + (x^2)f(x+1)

In this case, we would be looking for f(2). However, that equation doesn't contain enough information to solve it, because there are two free variables.

Any help would be appreciated.

 

[Millennial]

You found a functional equation. You only need one particular value of the function. For example, what is f(0)?
Note: You have an identity f(x)=f(-x) here. Use it.

 

[Bacle2]

Actually, this can be expressed as a recursion, indexed by a single term.Then

you can talk about convergence of the sequence:

You define a base term, and then every other term can be defined in terms of the previous one(s).

a1:=1

a2 is a function of a1.

Now, if you can find general properties of the sequence, you can ( when possible) determine its limit.