Fundamental Continuity Theorem

Given two Polynomials of the same order in one variable where the first $p$ Coefficients (but not the first $p-1$) are 0 and the Coefficients of the second approach the corresponding Coefficients of the first as limits, then the second Polynomial will have exactly $p$ roots that increase indefinitely. Furthermore, exactly $k$ Roots of the second will approach each Root of multiplicity $k$ of the first as a limit.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 4, 1959.