Limit laws and one-sided limits
Once you know a limit exists, a short list of algebra rules lets you compute almost any limit without touching the definition.
The limit laws
Limits respect the usual algebra: the limit of a sum is the sum of the limits, and the same holds for differences, products, and quotients as long as the denominator's limit is not zero. Constants pull outside, and powers and roots pass through. These laws let you evaluate limits of polynomials essentially by plugging in.
\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)One-sided limits
A one-sided limit only watches x approach a from the left (values below a) or from the right (values above a). The full limit exists exactly when both one-sided limits exist and agree. Piecewise-defined functions are the main place this check matters, because the formula changes at the boundary.
\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = LWorked example
Evaluate the limit of (x^2 - 9)/(x - 3) as x approaches 3. Plugging in gives 0/0, so factor first: the top is (x - 3)(x + 3). Canceling the common factor leaves x + 3, which is continuous, so the limit is 3 + 3 = 6.
\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} (x + 3) = 6Flashcard
Quotient law, valid when denominator limit is nonzero
Check your understanding
Let f(x) = x + 1 for x < 2 and f(x) = (x - 2)^2 + 3 for x at least 2. Does the limit of f(x) as x approaches 2 exist?