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1818 V. ANDRIEU, L. PRALY, AND A. ASTOLFInot uniquely defined. Indeed, if φ is homogeneous in the 0- (resp., ∞-) limit withassociated triple (r 0 ,d 0 ,φ 0 ) (resp., (r ∞ ,d ∞ ,φ ∞ )), then it is also homogeneous in the0- (resp., ∞-) limit with associated triple (kr 0 , k d 0 ,φ 0 ) (resp., (kr ∞ , k d ∞ ,φ ∞ )) forall k>0. (Simply change λ into λ k .)It is straightforward to show that if φ and ζ are two functions homogeneous inthe 0- (resp., ∞-) limit, with weights r φ,0 and r ζ,0 (resp., r φ,∞ and r ζ,∞ ), degrees d φ,0and d ζ,0 (resp., d φ,∞ and d ζ,∞ ), and approximating functions φ 0 and ζ 0 (resp., φ ∞and ζ ∞ ), then the following hold:P1: If there exists k in R + such that kr φ,0 = r ζ,0 (resp., kr φ,∞ = r ζ,∞ ), thenthe function x ↦→ φ(x) ζ(x) is homogeneous in the 0- (resp., ∞-) limit withweight r ζ,0 , degree kd φ,0 +d ζ,0 (resp., r ζ,∞ , kd φ,∞ +d ζ,∞ ) and approximatingfunction x ↦→ φ 0 (x) ζ 0 (x) (resp., x ↦→ φ ∞ (x) ζ ∞ (x)).dP2: If, for each j in {1,...,n},φ,0r φ,0,j< d ζ,0dr ζ,0,j(resp.,φ,∞r φ,∞,j> d ζ,∞r ζ,∞,j), thenthe function x ↦→ φ(x) +ζ(x) is homogeneous in the 0- (resp., ∞-) limitwith degree d φ,0 and weight r φ,0 (resp., d φ,∞ and r φ,∞ ) and approximatingfunction x ↦→ φ 0 (x) (resp., x ↦→ φ ∞ (x)). In this case we say that the functionφ dominates the function ζ in the 0-limit (resp., in the ∞-limit).P3: If the function φ 0 + ζ 0 (resp., φ ∞ + ζ ∞ ) is not identically zero and, for eachd φ,0r φ,0,j= d ζ,0r ζ,0,jdj in {1,...,n},(resp.,φ,∞r φ,∞,j= d ζ,∞r ζ,∞,j), then the functionx ↦→ φ(x) +ζ(x) is homogeneous in the 0- (resp., ∞-) limit with degreed φ,0 and weight r φ,0 (resp., d φ,∞ and r φ,∞ ) and approximating function x ↦→φ 0 (x) +ζ 0 (x) (resp., x ↦→ φ ∞ (x)+ζ ∞ (x)).Some properties of the composition or inverse of functions are given in the followingtwo propositions, the proofs of which are given in Appendices A and B.Proposition 2.10 (composition function). If φ : R n → R and ζ : R → R arehomogeneous in the 0- (resp., ∞-) limit functions, with weights r φ,0 and r ζ,0 (resp.,r φ,∞ and r ζ,∞ ), degrees d φ,0 > 0 and d ζ,0 ≥ 0 (resp., d φ,∞ > 0 and d ζ,∞ ≥ 0),and approximating functions φ 0 and ζ 0 (resp., φ ∞ and ζ ∞ ), then ζ ◦ φ is homogeneousin the 0- (resp., ∞-) limit with weight r φ,0 (resp., r φ,∞ ), degree d ζ,0 d φ,0r ζ,0(resp.,d ζ,∞ d φ,∞r ζ,∞), and approximating function ζ 0 ◦ φ 0 (resp., ζ ∞ ◦ φ ∞ ).Proposition 2.11 (inverse function). Let φ : R → R be a bijective homogeneousin the 0- (resp., ∞-) limit function with associated triple ( )1,d 0 ,ϕ 0 x d0 with ϕ0 ≠0and d 0 > 0 (resp., ( )1,d ∞ ,ϕ ∞ x d∞ with ϕ∞ ≠0and d ∞ > 0). Then the inversefunction φ −1 : R → R is a homogeneous in the 0- (resp., ∞-) limit function withassociated triple (1, 1 d 0, ( x ϕ 0) 1d 01) (resp., (1,d ∞, ( xϕ ∞) 1d∞ )).Despite the existence of well-known results concerning the derivative of a homogeneousfunction, it is not possible to say anything, in general, when dealing withhomogeneity in the limit. For example, the functionφ(x) =x 3 + x 2 sin(x 2 )+x 3 sin(1/x)+x 2 , x ∈ R ,is homogeneous in the bi-limit with associated triples(1, 2,x2 ) (, 1, 3,x3 ) .However, its derivative is homogeneous in neither the 0-limit nor the ∞-limit. Neverthelessthe following result holds, the proof of which is elementary.Proposition 2.12 (integral function). If the function φ : R n → R is homogeneousin the 0- (resp., ∞-) limit with associated triple (r 0 ,d 0 ,φ 0 ) (resp., (r ∞ ,d ∞ ,φ ∞ )),Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.