Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
Limit Laws are a set of rules that allow the calculation of limits for functions based on the limits of their constituent parts. They simplify the process of finding limits by breaking down complex expressions into simpler components.
5 Must Know Facts For Your Next Test
The Sum Law states that the limit of a sum is equal to the sum of the limits: $\lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x)$.
The Difference Law can be expressed as $\lim_{{x \to c}} [f(x) - g(x)] = \lim_{{x \to c}} f(x) - \lim_{{x \to c}} g(x)$.
The Constant Multiple Law indicates that $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot \lim_{{x \to c}} f(x)$, where $k$ is a constant.
The Product Law states that $\lim_{{x \to c}} [f(x) \cdot g(x)] = (\lim_{{x \to c}} f(x)) \cdot (\lim_{{x \to c}} g(x))$.
The Quotient Law asserts that if $\lim_{{x \to c}} g(x) \neq 0$, then $\lim_{{x \to c}} [\frac{f(x)}{g(x)}] = \frac{\lim_{{x \to c}} f(x)}{\lim_{{x \to c}} g(x)}$.
A function is continuous at point $c$ if $\lim_{x ->c} f(x) = f(c)$ and it is defined at that point.
Indeterminate Forms: Expressions like $\frac{0}{0}$ or $\infty - \infty$ which do not have well-defined values and require special techniques to evaluate.
$\epsilon-\delta$ Definition: $\epsilon-\delta$ definition formalizes the concept of a limit, stating that for every $\epsilon >0$, there exists a $\delta >0$ such that if $0<| x-c |<\delta$, then $|f ( x ) -L |<\epsilon$.