1.6 Determining Limits Using Algebraic Manipulation

Why algebraic manipulation?

Direct substitution often gives an indeterminate form like 0/0. When this happens, rewrite the expression using factoring, expanding, rationalizing (multiplying by the conjugate), or combining fractions with a common denominator so the problematic term cancels. After simplifying, substitute again to find the limit.

Factoring technique

For rational functions with 0/0 at x = a, factor numerator and denominator to find common factors, usually (x − a). Canceling removes the discontinuity that caused the indeterminate form, revealing the actual limiting behavior. This works because the limit only cares about values *near* a, not at a.

Rationalizing technique

When a limit involves a square root and gives 0/0, multiply numerator and denominator by the conjugate of the radical expression. This eliminates the root using the difference of squares, often revealing a cancelable factor. Simplify fully before substituting the limiting value.

Key terms

Indeterminate form

An expression like 0/0 that doesn't directly reveal the limit's value and requires further manipulation.

Factoring technique

Factor numerator/denominator to cancel common factors causing 0/0, then substitute.

Rationalizing technique

Multiply by the conjugate to eliminate square roots using difference of squares.

Common denominator technique

Combine fractions within an expression into one fraction to simplify before taking the limit.

Why cancellation works for limits

Limits depend on function behavior near a point, not at it, so removing a removable discontinuity doesn't change the limit.

Try an AP-style question

lim(x→3) (x² − 9)/(x − 3) = ?

1. 6
2. 0
3. Does not exist
4. 3