Limit Laws & Algebraic Techniques
Most limits of 'nice' functions can be found by direct substitution. When substitution gives an indeterminate form like 0/0, factor, rationalize, or simplify first.
Direct substitution first
For polynomials, rational functions (where the denominator isn't zero), and most common functions, lim(x→c) f(x) = f(c). Always try plugging in c first — it's the fastest path when it works.
Limit laws let you break a complicated limit into pieces: the limit of a sum/difference/product/quotient of functions is the sum/difference/product/quotient of their limits (as long as, for quotients, the denominator's limit isn't 0).
When substitution gives 0/0
A 0/0 result means substitution failed to reveal the answer — it does NOT mean the limit doesn't exist. It means there's a common factor (or a hidden cancellation) you need to expose first.
Standard tools: factor and cancel a common term, multiply by a conjugate to rationalize a radical, or combine fractions over a common denominator. After simplifying, substitute again.
Worked Example
Find lim(x→3) (x² - 9)/(x - 3).
- Direct substitution gives (9-9)/(3-3) = 0/0 — indeterminate, so factor.
- x² - 9 = (x-3)(x+3), so the expression simplifies to (x-3)(x+3)/(x-3) = x+3 for x ≠ 3.
- Now substitute: lim(x→3) (x+3) = 6.
Answer: 6
Key terms (1)
- Indeterminate form
- An expression like 0/0 that gives no direct information about the limit's value and signals more algebra is needed.
Practice Quiz
Find lim(x→2) (x² - 4)/(x - 2).