Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 2: Problem 2

Give the three conditions that must be satisfied by a function to be continuous at a point.

Short Answer

Expert verified
Answer: For a function to be continuous at a point, the following three conditions must be satisfied: 1. The function is defined at the point. 2. The limit of the function exists at the point. 3. The value of the function at the point equals the limit.

Step by step solution

01

Condition 1: Function is Defined at the Point

The function must be defined at the point in question. This means that the point should lie within the domain of the function.
02

Condition 2: Limit Exists at the Point

The limit of the function must exist at the point in question. In mathematical terms, the left-hand limit and the right-hand limit should both exist and be equal at the point. In symbols, for a function f(x) and a point c, \(\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)\).
03

Condition 3: Function Value Equals the Limit

The value of the function at the point should be equal to the limit. In mathematical terms, \(f(c) = \lim_{x \to c} f(x)\). This means that as x approaches the point c, the value of f(x) converges to the value of f(c).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. $$\begin{array}{l} g(1)=0, g(2)=1, g(3)=-2, \lim _{x \rightarrow 2} g(x)=0 \\ \lim _{x \rightarrow 3^{-}} g(x)=-1, \lim _{x \rightarrow 3^{+}} g(x)=-2 \end{array}$$

Let \(f(x)=\frac{2 e^{x}+5 e^{3 x}}{e^{2 x}-e^{3 x}} .\) Analyze \(\lim _{x \rightarrow 0^{-}} f(x), \lim _{x \rightarrow 0^{+}} f(x), \lim _{x \rightarrow-\infty} f(x)\) and \(\lim _{x \rightarrow \infty} f(x) .\) Then give the horizontal and vertical asymptotes of \(f .\) Plot \(f\) to verify your results.

Let \(f(x)=\frac{x^{2}-7 x+12}{x-a}\) a. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)\) equal a finite number? b. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)=\infty ?\) c. For what values of \(a\), if any, does \(\lim _{x \rightarrow a^{+}} f(x)=-\infty ?\)

Prove the following statements to establish the fact that \(\lim f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\). a. If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\) b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)

a. Sketch the graph of a function that is not continuous at 1, but is defined at 1. b. Sketch the graph of a function that is not continuous at 1, but has a limit at 1.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free