# The FCF (Functional Continued Fraction) #cosh_(cf) (x;a) = cosh(x+a/cosh(x+a/cosh(x+...)))#. How do you prove that #cosh_(cf) (0;1) = 1.3071725#, nearly and the derivative #(cosh_(cf) (x;1))'=0.56398085#, at x = 0?

See the explanation and the Socratic graph for y = cosh(x+1/y).

This FCF is produced by

At x=0,

y is equal to cosh(1/y).

Making use of the discrete analog iteration

8-sd approximation

y is equal to 1.3071725.

Now,

At x = 0,

Finding y' by substituting y = 1.3071725 and solving

Graph utilizing the inversion for y = cosh(x+1/y)

x=ln(y+(y^2-1)^0.5)-1/y} graph

The tangent at x = 0 is found on the second graph.

graph{(y-1.307-0.564x)=0}(x-ln(y+(y^2-1)^0.5)+1/y)

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To prove that cosh_(cf) (0;1) = 1.3071725, we can use the definition of the functional continued fraction and evaluate it numerically. By substituting x = 0 and a = 1 into the formula, we can calculate the value of the functional continued fraction.

To find the derivative (cosh_(cf) (x;1))' at x = 0, we can differentiate the functional continued fraction with respect to x and then substitute x = 0 into the derivative expression. By evaluating this expression numerically, we can find the value of the derivative at x = 0.

Please note that the numerical values provided (1.3071725 and 0.56398085) are specific approximations and may vary depending on the level of precision used in the calculations.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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