Limit laws

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

1. The Sum Law states that the limit of a sum is equal to the sum of the limits: $\lim_{{x \to c}} [f(x) + g(x)] = \lim_{{x \to c}} f(x) + \lim_{{x \to c}} g(x)$.
2. The Difference Law can be expressed as $\lim_{{x \to c}} [f(x) - g(x)] = \lim_{{x \to c}} f(x) - \lim_{{x \to c}} g(x)$.
3. 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.
4. The Product Law states that $\lim_{{x \to c}} [f(x) \cdot g(x)] = (\lim_{{x \to c}} f(x)) \cdot (\lim_{{x \to c}} g(x))$.
5. The Quotient Law asserts that if $\lim_{{x \to c}} g(x) \neq 0$, then $\lim_{{x \to c}} [\frac{f(x)}{g(x)}] = \frac{\lim_{{x \to c}} f(x)}{\lim_{{x \to c}} g(x)}$.

Review Questions

• What does the Sum Law state about limits?
• How do you apply the Constant Multiple Law when calculating a limit?
• Under what condition can you use the Quotient Law for limits?

Related terms

Continuity: 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 $|f ( x ) -L |<\epsilon$.