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 the remaining polynomial factor. If, after simplifying, one final numerator factor still doesn't cancel against a matching denominator factor, the two-sided limit does not exist — the function grows without bound near a, sometimes to plus infinity from one side and minus infinity from the other.