Using the results in pages 59-63 of Rosenblum and Rovnyak (p. 62 in particular, and noting that your Toeplitz matrices are symmetric, hence normal, so the operator norm is what you want), it follows that $\lim_n \lambda_n(x) = \frac{1-x^2}{(1-|x|)^2}$. The non-asymptotic behavior is probably hard, but numerics clearly indicate (and it is probably not hard to show) that $\lambda_n(1) = n$. It might be possible to combine these observations, or more detailed ones along similar lines, to get good estimates for $\lambda_n(x)$.

Using the results in pages 59-63 of Rosenblum and Rovnyak (p. 62 in particular, and noting that your Toeplitz matrices are symmetric, hence normal, so the operator norm is what you want), it follows that $\lim_n \lambda_n(x) = \frac{1+x}{1-x}$. The non-asymptotic behavior is probably hard, but numerics clearly indicate (and it is probably not hard to show) that $\lambda_n(1) = n$. It might be possible to combine these observations, or more detailed ones along similar lines, to get good estimates for $\lambda_n(x)$.