Consider a tridiagonal $n\times n$ matrix with the following format 

$$ \mathbf A = \begin{bmatrix}
a & b & & \phantom{\ddots} & \phantom{\ddots} \\
b & a & b & \phantom{\ddots} & \\
 & b & a & \ddots & \\
 & & \ddots & \ddots & b \\
\phantom{\ddots} & \phantom{\ddots} & \phantom{\ddots} & b & a \\
\end{bmatrix}.$$

This matrix has eigenvalues equal to $$\lambda\_k = a-2b\cos\left(\frac{k\pi}{1+n}\right),\;\text{ for }\;k\in\{1,\dots,n\}.$$ It follows that $$\lim\_{n\to\infty}\lambda\_k=a-2b.$$

It is also symmetric which mean it can be diagonalized into $$\mathbf A= \mathbf Q \mathbf \Lambda \mathbf Q^T,\;\text{ so that }\;\mathbf A^{1/2}= \mathbf Q \mathbf \Lambda^{1/2} \mathbf Q^T.$$ 

Is it possible to obtain an analitical formula for the elements of $\lim\_{n\to\infty}\mathbf A^{1/2}$?

**Edit:** [This paper on tridiagonal Toeplitz matrices appears to be helpful](http://www.math.kent.edu/~reichel/publications/toep3.pdf).