In this paper, Newman-type rational interpolation to some limited smooth functions induced by some set of nodes X in [-1, 1] is considered and the upper and lower bound of the order of approximation is given, which is sharper than Stahl's result. And the error estimation is Cn(-alpha 2/alpha+1) exp(-n(alpha/alpha+1)) <= vertical bar e(n)(X:x)vertical bar <= 3 exp(-n(alpha/alpha+1)). (C) 2016 Elsevier GmbH. All rights reserved.