Determining Limits Using Algebraic Properties

A limit asks what value a function closes in on as x approaches some number. The limit laws say you can split a complicated expression into simple pieces, find the limit of each piece, then recombine: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits, as long as the bottom one isn't zero. Polynomials are built entirely from additions and multiplications of x, so applying these rules piece by piece turns out to be exactly the same as just plugging the number in. That is why direct substitution works for polynomials and most rational functions: it's the limit laws running automatically in the background.

The quotient law has one condition that trips people up: it only applies when the limit of the denominator is not zero. If substitution gives you a nonzero number over zero, the limit does not exist, usually because the function shoots off to infinity. If substitution gives you 0/0, that is not an answer at all, it's a signal that the quotient law cannot be used yet. The function might still have a perfectly good limit, but you have to factor, cancel, rationalize, or otherwise rewrite the expression so the problematic factor disappears before substituting again.

This topic is the toolkit that makes every later limit technique possible. Once a function is continuous at the point in question, these laws justify replacing a limit with a plain evaluation, which is the bridge into continuity itself. When substitution fails because of a 0/0 form, that failure is also the opening move for factoring techniques and, eventually, for L'Hopital's rule once derivatives are available.