Q.$\displaystyle\lim_{x\to 2}\left[\frac{x^3-4x^2+4x}{x^2-1}\right]$

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 …