A limit lim[x→a] f(x) tells us where f(x) is heading as x gets close to a. When substituting x = a gives 0/0, the function almost always has a factor we can cancel.
The limit of f(x) as x approaches aRemovable discontinuity
A hole at x = a where the limit exists but the function is undefined. For (x² − a²)/(x − a) the numerator factors to (x − a)(x + a), so the quotient simplifies to x + a and the limit is 2a.
The standard limit lim[x→0] sin(x)/x = 1 (x in radians) is the seed of the derivative of sine. By scaling, lim[x→0] sin(kx)/x = k.
Trick of the trade
Factor first, cancel second, substitute last. If cancellation is impossible, evaluate numerically from both sides — if the two sides disagree, the limit does not exist.