/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Describe the points (if any) at ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 6

Describe the points (if any) at which a rational function fails to be continuous.

Short Answer

Expert verified
Answer: A rational function fails to be continuous at the points where its denominator is equal to zero.

Step by step solution

01

Understand the concept of continuity

A function is continuous at a point if the following conditions are satisfied: 1. The function is defined at the point 2. The limit of the function as x approaches the point exists 3. The limit of the function as x approaches the point is equal to the function's value at that point. If any of these conditions are not met, the function is considered discontinuous at the point.
02

Write down the rational function

A rational function can be written as the ratio of two polynomial functions, i.e., f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions.
03

Identify conditions under which rational functions are discontinuous

A rational function is discontinuous when one or more conditions of continuity are not met. The most common reason for a rational function to be discontinuous is when the denominator Q(x) is equal to zero. This is because division by zero is undefined, which means that the function will not be defined at that point.
04

Find values of x for which the denominator Q(x) is equal to zero

To find the points at which the rational function is discontinuous, we need to find the values of x at which the denominator, Q(x), is equal to zero. To do this, set Q(x) equal to zero and solve for x using algebraic techniques. The solutions to Q(x) = 0 will be the points where the rational function is discontinuous.
05

Verify discontinuity

Lastly, we must verify that the points found in the previous step indeed make the rational function discontinuous. If the limit of the function as x approaches the point does not exist or is not equal to the function's value at that point, then the function is discontinuous at that point. To summarize, the points at which a rational function fails to be continuous are the solutions to Q(x) = 0, i.e., the values of x where the denominator of the rational function is equal to zero.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{3 x^{4}+3 x^{3}-36 x^{2}}{x^{4}-25 x^{2}+144}$$

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(\theta)=\tan \left(\frac{\pi \theta}{10}\right)$$

Refer to Exercises \(91-92.\) a. Does the function \(f(x)=x \sin (1 / x)\) have a removable discontinuity at \(x=0 ?\) b. Does the function \(g(x)=\sin (1 / x)\) have a removable discontinuity at \(x=0 ?\)

Determine whether the following statements are true and give an explanation or counterexample. a. The graph of a function can never cross one of its horizontal asymptotes. b. A rational function \(f\) can have both \(\lim _{x \rightarrow \infty} f(x)=L\) and \(\lim _{x \rightarrow-\infty} f(x)=\infty\). c. The graph of any function can have at most two horizontal asymptotes.

Let $$g(x)=\left\\{\begin{array}{ll}1 & \text { if } x \geq 0 \\\\-1 & \text { if } x<0\end{array}\right.$$ a. Write a formula for \(|g(x)|\) b. Is \(g\) continuous at \(x=0 ?\) Explain. c. Is \(|g|\) continuous at \(x=0 ?\) Explain. d. For any function \(f,\) if \(|f|\) is continuous at \(a,\) does it necessarily follow that \(f\) is continuous at \(a ?\) Explain.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.