Consider a polynomial equation in the standard form

c_nxⁿ + c_n-1x^n-1 + + c₁x + c₀ = 0

Consider the smallest magnitude root of this equation as c₀ → 0. That root would be proportional to c₀. For x very small the terms involving the higher powers of x become insignficant compared to c₁x and the polynomial equation reduces asymptotically to

c₁x + c₀ = 0
and thus
x = − c₀/c₁ = 0

Thus for small values of c₀, x=−c₀/c₀ regardless of the magnitude of the other coefficients

The degree of the polynomial may be infinite as well as finite.

A function f(x) has a Maclaurin series of the form

f(x) = f(0) + f'(0)x + ½f"(0)x² + …

Therefore the smallest root of f(x)=0 asymptotically approaches −f(0)/f'(0) as f(0)→0.