AcademyLimits
Academy
Limit Laws
Level 1 - Math I (Physics) topic page in Limits.
Limit Laws and Operations
When computing limits, we can use established laws that govern how limits interact with basic algebraic operations. These rules form the foundation for evaluating limits algebraically.
Basic Limit Laws
If \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\), then:
Sum Law:
Sum Rule
\[\lim_{x \to a} [f(x) + g(x)] = L + M = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\]
Difference Law:
Difference Rule
\[\lim_{x \to a} [f(x) - g(x)] = L - M = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)\]
Product Law:
Product Rule
\[\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\]
Quotient Law:
Quotient Rule
\[\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}, \text{ provided } M \neq 0\]
Constant Multiples
Constant Multiple
\[\lim_{x \to a} [c \cdot f(x)] = c \cdot L, \text{ where } c \text{ is a constant}\]
Power and Root Laws
Power Rule
\[\lim_{x \to a} [f(x)]^n = L^n, \text{ for any positive integer } n\]
Root Rule
\[\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}, \text{ provided } L > 0 \text{ when } n \text{ is even}\]
Applying the Laws
When using these laws, ensure:
- Each individual limit exists
- For quotients, the denominator limit is non-zero
- The laws apply to sums, differences, products, and quotients of any finite number of functions
These algebraic properties make it possible to evaluate limits of complex expressions by breaking them into simpler components whose limits we already know.