MATH 01130 · Limits & continuity
Limit laws, one-sided limits
Definition
Limit laws let you break lim(f±g), lim(cf), lim(fg), lim(f/g) (if lim g ≠ 0) into limits of pieces. One-sided limits lim x→a⁺ and lim x→a⁻ must both exist and match for the two-sided limit to exist.
Read this
Think of a limit as the y-value the function approaches as x approaches a — not necessarily f(a). Piecewise graphs and holes are classic.
Workflow: try direct substitution → simplify algebraically (factor, conjugate, cancel) → check one-sided values if there is a break or absolute value.
Common trap: lim f/g can fail even if lim f and lim g exist, when lim g = 0. Also lim |x|/x as x→0 has one-sided limits 1 and −1, so the two-sided limit DNE.
Drill with flashcards
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Try this
Attempt each problem first, then open the worked solution.
1.Compute lim x→2 (x²−4)/(x−2) by factoring.
2.Decide if lim x→0 x/|x| exists.