Q.$\displaystyle\lim_{y\to 0}\left[\frac{5y^3+8y^2}{3y^4-16y^2}\right]$

When a rational function's limit produces the indeterminate form zero over zero at x equals a, it is a certain sign that (x minus a) divides evenly into both the numerator polynomial and the denominator polynomial. The method of factorization makes this cancellation explicit: factor the numerator completely, factor the denominator completely, cancel every factor of (x minus a) that appears in both (this cancellation is valid because, on the way to the limit, x is never actually equal to a, so x minus a is never literally zero), and substitute x equals a into whatever remains. When the shared factor isn't obvious by eye, synthetic division against the suspected root a quickly produces …