Chapter 1.3, Problem 10ES

Suppose that as x approaches zero,

$F_1(x)=L_1+O\left(x^{\alpha }\right)\text{and}{F}_2(x)=L_2+O\left(x^{\beta }\right).$

Let c₁ and c₂ be nonzero constants and define

$F(x)=c_1{F}_1(x)+c_2{F}_2(x)\text{and}G(x)=F_1\left(c_{1}x\right)+F_2\left(c_{2}x\right).$

Show that if γ = minimum {α, β}, then, as x approaches zero,

1. a. F(x) = c₁L₁ + c₂L₂ + O(x^γ)
2. b. G(x) = L₁ + L₂ + O(x^γ).