Limits & Continuity

Evaluating Limits Algebraically

Using algebraic techniques (factoring, rationalizing) to evaluate limits.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Estimating Limits

Evaluating Limits Without a Table

A table can only ever estimate a limit. Before reading on: for limₓ→4 (x² − 16)/(x − 4), which also gives 0/0, what algebraic move would resolve it exactly, the way it did for the previous lesson's rational function?

Worked Example — Resolving 0/0 by Factoring

Evaluate limₓ→4 (x² − 16)/(x − 4). Direct substitution gives 0/0. Factor: (x − 4)(x + 4)/(x − 4) = x + 4, valid for x ≠ 4 — exactly what the limit cares about. So limₓ→4 (x² − 16)/(x − 4) = limₓ→4 (x + 4) = 8.

Worked Example — Resolving 0/0 by Rationalizing

Evaluate limₓ→0 (√(x + 16) − 4)/x. Direct substitution gives 0/0. Multiply by the conjugate, √(x+16) + 4: [(x+16) − 16] / [x(√(x+16) + 4)] = x / [x(√(x+16) + 4)] = 1/(√(x+16) + 4), valid for x ≠ 0. As x → 0: 1/(4+4) = 1/8. Check numerically: at x = 0.001, (√16.001 − 4)/0.001 ≈ 0.125, matching 1/8.

Tip

Try direct substitution first, always — reach for factoring or rationalizing only when substitution produces an indeterminate form like 0/0.

Common Mistakes

Substituting into the original, unsimplified expression after factoring or rationalizing.
After resolving an indeterminate form algebraically, substitute into the simplified expression — that's the whole point of the algebra.

Key Takeaways

A 0/0 result from direct substitution doesn't mean the limit fails to exist — factoring or rationalizing often resolves it.
Factoring works well when a common factor causes the 0/0; rationalizing works well when a square root does.

Summary

Algebraic techniques evaluate limits exactly. The next lesson uses limits to define precisely what it means for a function to be continuous.

Track your progress

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