Power law for limits

Definition

The power law for limits states that if the limit of a function $f(x)$ as $x$ approaches $c$ is $L$, then the limit of $[f(x)]^n$ as $x$ approaches $c$ is $L^n$, provided that n is a positive integer.

5 Must Know Facts For Your Next Test

1. The power law for limits applies to functions raised to any positive integer exponent.
2. If $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [f(x)]^n = L^n$.
3. This law helps simplify the evaluation of limits involving polynomial functions.
4. It requires that the initial limit $\lim_{{x \to c}} f(x)$ exists and is finite.
5. The power law can be combined with other limit laws to solve more complex problems.

Review Questions

Related terms

Limit:

The value that a function approaches as the input approaches some value.

Polynomial Function:

A mathematical expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.

Continuous Function: A function without breaks, jumps, or holes in its domain; it has a continuous graph.