Selecting Procedures for Determining Limits

No single algebra move cracks every limit, because the obstruction blocking direct substitution changes from problem to problem. Sometimes it's a factorable polynomial, sometimes a radical, sometimes a messy fraction inside a fraction, sometimes a trig expression that hides behind sin(x)/x. So limit-finding works like a decision tree: try plugging in first. If that gives a real number, you're done. If it gives 0/0, the function has a removable gap, and you look at its shape to decide the repair. A difference of squares or a common root signals factoring. A square root being added or subtracted signals multiplying by a conjugate. A fraction stacked over another fraction signals clearing denominators first. An expression built from sin(x) or (1 - cos(x)) over x signals the special trig limits, sin(x)/x → 1 and (1 - cos(x))/x → 0 as x → 0.

Each tool does the same job: rewrite the function so the troublesome factor cancels, revealing what the function approaches even though it never actually equals that value at the point itself. This is the heart of why limits exist — they describe behavior near a point, independent of whatever happens (or fails to happen) exactly at that point.

This topic sits right before continuity, where the question becomes whether the limit actually matches the function's value at that point. Selecting procedures well here also sets up derivatives, since every derivative is a limit of a difference quotient that starts life looking like 0/0.