1.5 Determining Limits Using Algebraic Properties of Limits

Limit Laws for Combining Functions

If lim f(x) and lim g(x) both exist as x→c, you can compute limits of combinations directly: sum/difference rule adds or subtracts the limits, product rule multiplies them, and quotient rule divides them (as long as the denominator's limit ≠ 0). Constant multiples pass through: lim[k·f(x)] = k·lim f(x). These properties let you break complex expressions into simpler pieces.

Direct Substitution First

For polynomial and rational functions, try direct substitution first: plug in c and evaluate. If the denominator isn't zero, this gives the limit immediately. This works because polynomials and rational functions are continuous wherever they're defined, so the limit equals the function value.

Handling Indeterminate Forms

If direct substitution gives 0/0, the function likely has a removable factor. Use algebraic techniques — factoring, rationalizing (multiplying by a conjugate), or simplifying complex fractions — to cancel the problematic term, then substitute again. This reveals the actual limit even though the original expression was undefined at that point.

Key terms

Sum Rule for Limits

lim[f(x) + g(x)] = lim f(x) + lim g(x), provided both limits exist.

Quotient Rule for Limits

lim[f(x)/g(x)] = lim f(x) / lim g(x), provided lim g(x) ≠ 0.

Indeterminate form 0/0

A result from direct substitution signaling a common factor exists; simplify algebraically (factor/rationalize) then re-substitute.

Direct substitution

Plugging the x-value directly into the function; works for polynomials and rational functions where the denominator isn't zero.

Rationalizing technique

Multiplying numerator and denominator by the conjugate to eliminate a radical, useful for limits with square roots.

Try an AP-style question

If lim(x→a) f(x) = 3 and lim(x→a) g(x) = −2, what is lim(x→a) [2f(x) − (g(x))²]?

1. 10
2. 2
3. 8
4. −10