1.7 Selecting Procedures for Determining Limits

Choosing the right tool

When finding a limit, first try direct substitution. If that gives a real number, you're done. If it gives an indeterminate form like 0/0, try factoring and canceling, rationalizing (multiplying by conjugates), simplifying complex fractions, or trigonometric identities before substituting again.

Watch for special forms

Some limits require known results: lim(x→0) sin(x)/x = 1 and lim(x→0) (1−cos(x))/x = 0. If substitution gives a nonzero number over zero, the limit may be infinite (check one-sided limits and signs) — a sign of a vertical asymptote, not a removable hole.

Piecewise and multi-step limits

For piecewise functions, check the left-hand and right-hand limits separately at the boundary point; they must match for the two-sided limit to exist. Sometimes multiple algebraic techniques must be combined — e.g., factor first, then rationalize — so always simplify step-by-step before re-substituting.

Key terms

First strategy for evaluating limits

Try direct substitution first; if it produces a real number, that is the limit.

Indeterminate form 0/0 — next steps

Try factoring and canceling, rationalizing (conjugates), simplifying complex fractions, or using trig identities.

lim(x→0) sin(x)/x

Equals 1 — a key special trigonometric limit.

lim(x→0) (1−cos(x))/x

Equals 0 — another standard trigonometric limit.

Nonzero constant over 0 after substitution

Indicates the limit is likely infinite (vertical asymptote); check one-sided limits and signs.

Try an AP-style question

lim(x→0) sin(3x) / x = ?

1. 1/3
2. 1
3. 3
4. 0