We prove that if $P(x)$ is a tame polynomial of several variables and if $Q(x)$ is a polynomial of degree lower than one plus the Lojasiewicz number at infinity of $P(x)$, then the Milnor fibrations at infinity of $P(x)$ and $P(x)+Q(x)$ are equivalent. This fact can be considered as a version at infinity of the Kuiper-Kuo theorem. We also relate the  Lojasiewicz number at infinity to the phenomenon of singularities at infinity.