Limits Examples

Calculus Readiness · Axiom Academy

EXAMPLE Computing Limits Algebraically Worked examples: direct substitution, factor-and-cancel, the conjugate trick, one-sided limits, limits at infinity, and the key special limit. Most limits come down to a short checklist. Work through the nine examples below and decide, each time, which move applies. Try direct substitution first. If you get a number, you are done. If you get (an indeterminate form), reach for factor-and-cancel, the conjugate, or combining fractions. If you get , the limit is infinite — it is a vertical asymptote, so check the sign. Nice work — you just ran the whole limit toolkit on nine problems. Almost every algebraic limit reduces to one of these moves. Substitute first: if a function is continuous at the point (every polynomial is), the limit is simply its value. means "try harder": factor and cancel, multiply by the conjugate, or combine a complex fraction to remove the trouble factor, then substitute. is an asymptote: the limit is — decide the sign from the one-sided behavior, not by factoring. At infinity, the highest power wins: divide every term by it; equal degrees give the ratio of leading coefficients. Know the special limit: — reshape the expression until the angle matches the denominator. When an indeterminate form appears, the only question is: which trick unglues this one?

This is the written version of the interactive lesson above. See the full Calculus Readiness course.