Describe the intervals on which each function f in exercises...

Name: Problem 73, Chapter 1: Calculus: Early Transcendentals
Uploaded: 2022-02-23T14:19:43-08:00
Duration: 1 min 21 s
Channel: Lauren Shelton
Description: Describe the intervals on which each function f in exercises is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity and any left-or right-continuity. f(x)={ x^2-3 x-1, if x ≠-2 3, if x=-2 .

Describe the intervals on which each function $f$ in exercises is continuous. At each point where $f$ fails to be continuous, use limits to determine the type of discontinuity and any left-or right-continuity. $$ f(x)=\left\{\begin{aligned} x^{2}-3 x-1, & \text { if } x \neq-2 \\ 3, & \text { if } x=-2 \end{aligned}\right. $$

Describe the intervals on which each function $f$ in exercises is continuous. At each point where $f$ fails to be continuous, use limits to determine the type of discontinuity and any left-or right-continuity.
$$
f(x)=\left\{\begin{aligned}
x^{2}-3 x-1, & \text { if } x \neq-2 \\
3, & \text { if } x=-2
\end{aligned}\right.
$$

Key Concepts

Continuity of Functions

This concept refers to the property of a function such that small changes in the input lead to small changes in the output. In formal terms, a function is continuous at a point if the limit of the function as the input approaches that point equals the function’s value at that point. Understanding continuity is fundamental to calculus and analysis as it underpins many important theorems and applications.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific point from one direction only (either from the left, denoted as the left-hand limit, or from the right, denoted as the right-hand limit). They are crucial in analyzing the behavior of functions at points of discontinuity and in classifying the type of discontinuity present.

Piecewise Functions

These are functions defined by different expressions over different intervals of the domain. Analyzing piecewise functions involves checking continuity at the points where the definition changes, typically by computing one-sided limits and ensuring they match the function's assigned value at those points.

Polynomial Functions

Polynomial functions are continuous over the entire real line because they are composed of sums, differences, and products of continuous power functions. This inherent continuity makes them useful when they are used as components of more complex, piecewise-defined functions.

Removable Discontinuity

A removable discontinuity occurs at a point where a function is not continuous, yet the limit exists as the input approaches that point. This type of discontinuity hints that the function’s definition at that point could be modified to make the function continuous, often by redefining the function value to equal the limit.

Transcript

Please give Ace some feedback