A calculus problem by Hobart Pao

If the linear approximation of $\sqrt{2.2+\sqrt{2.1+\sqrt{2.2+\sqrt{2.1+\sqrt{2.2+\cdots}}}}}$ , using the function $z= \sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\sqrt{x+\cdots}}}}}$ and initial point $(2,2,2)$ , is $L$ , find $100L$ .

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Note: The linear approximation to a function $z=f(x,y)$ at $(x, y)$ is $f(x, y) \approx f(a,b) + f_x (a, b) (x-a) + f_y (a, b) (y-b)$ , where $(a, b, f(a, b) )$ is the initial point. This comes from the equation of a tangent plane, and we are using a tangent plane to find a linear approximation.