FCF (Functional Continued Fraction) #cosh_(cf) (x; a)=cosh(x+a/cosh(x+a/cosh(x+...)))# How do you prove that #y = cosh_(cf) (x; x)# is asymptotic to #y = cosh x#, as #x -> 0# or the graphs touch each other, at #x = 0#?
The Socratic graphs touch at x = 0 and the point of contact is (0, 1).
At x = 0, for
Like cosh x, this is also an even function of x.
At (0, 1),
For y = cosh x also, when x = 0, y=1 and y' = sinh x = 0, at (0, 1)
Thus, both touch each other at (0, 1), with cosh_(cf) graph bracing
cosh graph, from above.
The equation of the common tangent is y = 1
Graph of y = cosh x:
Graph of the FCF y = cosh(x+x/y):
Combined Socratic graph for y = cosh x and the FCF y =
and the common tangent y = 1: