# What is the graphic of #f(x) = sqrt(x+sqrt(x+sqrt(x+sqrt(x+...))))# for #x ge 0#?

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What is the graphic of #f(x) = sqrt(x+sqrt(x+sqrt(x+sqrt(x+...))))# for #x ge 0# ?

What is the graphic of

##### 1 Answer

This is the continued-surd model for the equation of part of a parabola, in the first quadrant. Not in the graph, the vertex is at #(-1/4, 1.2) and the focus is at (0, 1/2).

#### Explanation:

As of now,

The graph is the part of a parabola that has vertex at

and latus rectum 4a = 1.. The focus is at

As

quadrant, wherein

I think it is better to restrict x as > 0, to avoid (0, 1) of the parabola.

Unlike parabola y, our y is single-valued, with

graph{(x+y-y^2)((x-4)^2+(y-2.56)^2-.001)=0[0.1 5 1 5] }

I make it for another g in continued-surd

Let g(x) = ln x. Then

Here,

graph{((ln x+y)^0.5-y)((x-1)^2+(y-1)^2-.01)=0[0..779 1 0.1 1] }