Let \(K(a_ n,1;x_ 1)\) be a limit-periodic modified continued fraction with \(\lim_{n\to \infty}a_ n=a\in {\mathbb{C}}-(-\infty,1/4)\) and n-th approximant \[ g_ n=S_ n(x_ 1)=a_ 1/1+a_ 2/1+...+a_{n- 1}/1+a_ n/(1+x_ 1), \] where \(x_ 1\) denotes the smaller (in modulus) of the two fixed points of \(T(w)=a/(1+w).\) Further, let \(f_ n=S_ n(0)\) be the n-th ordinary reference continued fraction \(K(a_ n/1)\). The authors give truncation error bounds for both \(g_ n\) and \(f_ n\) and show that certain a posteriori bounds for \(g_ n\) are the best possible. The paper also includes results on the speed of convergence and applications to a number of special functions. Very interesting numerical examples indicate the sharpness of the error bounds.