Root law for limits

Definition

The Root Law for Limits states that the limit of a root of a function is equal to the root of the limit of that function, provided the initial limit exists and is within the domain of the root function. Mathematically, if $\lim_{{x \to c}} f(x) = L$ and $n$ is a positive integer, then $\lim_{{x \to c}} \sqrt[n]{{f(x)}} = \sqrt[n]{{L}}.$

5 Must Know Facts For Your Next Test

1. The Root Law for Limits requires that the limit $L$ must exist.
2. The value under the root must be non-negative if $n$ is an even integer.
3. This law can be applied iteratively or in combination with other limit laws.
4. It simplifies complex limit problems involving roots by breaking them down into simpler components.
5. Understanding this law is crucial for solving higher-level calculus problems involving limits.

Review Questions

Related terms

Limit:

The value that a function (or sequence) approaches as the input (or index) approaches some value.

Continuity:

A function is continuous at a point if it is defined at that point, its limit exists at that point, and its limit equals its value at that point.

$\epsilon-\delta$ Definition: $\epsilon-\delta$ definition provides a formal description of a limit, stating that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$.