1.7 Selecting Procedures for Limits

A strategy

1) Try direct substitution first. 2) If you get a finite number (continuous function), that's the answer. 3) If you get 0/0, simplify (factor / rationalize / combine).

Non-zero over zero

If substitution gives a (nonzero number)/0, the limit is infinite — expect a vertical asymptote, and check one-sided behavior. Pick the most efficient method for the form you get.

Key terms

Substitution first

Always try plugging in c before anything else.

0/0 → simplify

Factor, rationalize, or combine, then substitute.

(nonzero)/0

Signals an infinite limit / vertical asymptote.

Choosing a method

Match the technique to the form substitution produces.

Finite result

For continuous f, the substituted value is the limit.

Try an AP-style question

lim(x→0) sin(3x) / x = ?

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