Selecting Procedures for Determining Limits AB & BC

With every technique in the unit available, the work becomes choosing among them. The routine is fixed: substitute first, then branch on the result. A determinate number is the answer. A $\frac{0}{0}$ calls for a rewrite chosen by structure, factoring for polynomials, a conjugate for radicals, a common denominator for complex fractions. A nonzero over zero means unbounded behavior and no limit. A piecewise or absolute-value join calls for one-sided limits before anything else.

Three errors recur. Substituting and reporting whatever appears, so an unfinished $\frac{0}{0}$ becomes the answer. Treating that form as a value or as proof of failure, when most such limits resolve to a finite number. And applying a limit law whose hypotheses fail, most often splitting a quotient whose denominator tends to zero, or splitting a product one of whose factors has no limit at all.