Suppose that as x approaches zero, F1(x) = L1 + O(x) and F2(x) = L2 + O(x).

Question:

Suppose that as x approaches zero,
F1(x) = L1 + O(xα) and F2(x) = L2 + O(xβ).
Let c1 and c2 be nonzero constants, and define
F(x) = c1F1(x) + c2F2(x) and
G(x) = F1(c1x) + F2(c2x).
Show that if γ = minimum {α, β}, then as x approaches zero,
a. F(x) = c1L1 + c2L2 + O(xγ)
b. G(x) = L1 + L2 + O(xγ).