Limit Laws

Limit Laws: Combining Simple Limits

Limit laws are algebraic rules that let you evaluate limits of complicated expressions by breaking them into simpler pieces. If $\lim_{x\to a} f(x) = L$ and $\lim_{x\to a} g(x) = M$ both exist, you can combine these limits using predictable rules—without graphing or building tables.

Core Rules

The sum, difference, and product rules work intuitively:

$\lim_{x\to a}[f(x)+g(x)] = L+M \quad \lim_{x\to a}[f(x)-g(x)] = L-M \quad \lim_{x\to a}[f(x)\cdot g(x)] = L\cdot M$

A special case is the constant multiple rule: $\lim_{x\to a} c\,f(x) = c\,L$.

The quotient rule requires one critical condition:

$\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{L}{M} \quad \text{provided } M \neq 0$

For powers and roots:

$\lim_{x\to a} [f(x)]^n = L^n \quad \lim_{x\to a} \sqrt[n]{f(x)} = \sqrt[n]{L}$

Direct Substitution

These laws justify direct substitution: for polynomials, rational functions (where the denominator isn't zero), and root functions at domain points, simply plug in the value. For example, $\lim_{x\to 2}\frac{3x^2-5}{x+1} = \frac{3(4)-5}{3} = \frac{7}{3}$.

When Limit Laws Break Down

The quotient rule fails when $M = 0$. Division by zero creates an indeterminate form ($0/0$) or undefined expression. When this occurs, you need algebraic techniques like factoring or rationalizing to resolve the limit—the rule simply doesn't apply.

Example

Evaluate $\lim_{x\to 1}\sqrt{\frac{2x^3+4}{x+3}}$:

$\lim_{x\to 1}(2x^3+4) = 6, \quad \lim_{x\to 1}(x+3) = 4 \;(\neq 0)$

$\Rightarrow \lim_{x\to 1}\frac{2x^3+4}{x+3} = \frac{6}{4} = \frac{3}{2} \quad \Rightarrow \quad \lim_{x\to 1}\sqrt{\frac{2x^3+4}{x+3}} = \sqrt{\frac{3}{2}}$

Recognizing when limit laws apply—and when they fail—is as important as knowing the laws themselves.