Abstract: A new class of positive linear operators is introduced,and the simultaneous approximation of unbounded functions is studied.Let ｆ∈Ｃβ[０，∞），ｒ∈Ｎ，derivative of ordern n exists in ｆ（ｘ） in [０，∞）,then ｌｉｍｎ∞Ｍ（ｒ）ｎ，α（ｆ（ｔ），ｘ）＝ｆ（ｒ）（ｘ），if ｆ（ｒ）（ｘ）∈Ｃ（ａ－η，ｂ＋η）（η＞０），then Ｍ（ｒ）ｎ，α（ｆ，ｘ）ｆ（ｒ）（ｘ） is applicable when ｘ∈[ａ，ｂ].If ｆ∈Ｃβ[０，∞），ｆ（ｘ） has derivative of order ｒ＋２ in [０，∞），then ｌｉｍｎ∞ｎ[Ｍ（ｒ）ｎ，α（ｆ，ｘ）－ｆ（ｒ）（ｘ）]＝α[ｒ（ｒ＋１）ｆ（ｒ）（ｘ）＋（２（ｒ＋１）ｘ＋ｒ）ｆ（ｒ＋１）（ｘ）＋ｘ（１＋ｘ）ｆ（ｒ＋２）（ｘ）],if ｆ（ｒ＋２）（ｘ）∈Ｃａ－η，ｂ＋η）（η＞０），then the above equation is valid in [ａ，ｂ].

Key words: simultaneous approximation, modulus of smoothness, asymptotic expansion ;approximation degree