Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.6.82

Chapter 2, Problem 2.6.82

Suppose that f(x) and g(x) are polynomials in x. Can the graph of f(x)/g(x) have an asymptote if g(x) is never zero? Give reasons for your answer.

Verified step by step guidance

To determine if the graph of f(x)/g(x) can have an asymptote, we need to understand what an asymptote is. An asymptote is a line that the graph of a function approaches but never touches or crosses.

There are two main types of asymptotes for rational functions: vertical and horizontal (or oblique). Vertical asymptotes occur where the denominator of the function is zero, causing the function to be undefined at that point.

Since g(x) is never zero, there cannot be any vertical asymptotes for the function f(x)/g(x). Vertical asymptotes are directly related to the points where the denominator equals zero, which is not the case here.

Next, consider horizontal or oblique asymptotes. These occur based on the degrees of the polynomials f(x) and g(x). If the degree of f(x) is less than the degree of g(x), the horizontal asymptote is y=0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

Therefore, even if g(x) is never zero, the graph of f(x)/g(x) can still have a horizontal or oblique asymptote, depending on the relative degrees of f(x) and g(x).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomials

Polynomials are mathematical expressions consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. They can be represented in the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n are coefficients and n is a non-negative integer. Understanding polynomials is crucial for analyzing their behavior, including their graphs and potential asymptotic behavior.

Asymptotes

Asymptotes are lines that a graph approaches but never touches or intersects. They can be vertical, horizontal, or oblique, and they indicate the behavior of a function as it approaches certain values or infinity. In the context of rational functions like f(x)/g(x), vertical asymptotes occur where g(x) is zero, while horizontal asymptotes relate to the degrees of the polynomials involved.

Rational Functions

Rational functions are ratios of two polynomials, expressed as f(x)/g(x). The behavior of these functions, particularly their asymptotic behavior, is influenced by the degrees of the numerator and denominator polynomials. If g(x) is never zero, the function does not have vertical asymptotes, but it can still exhibit horizontal asymptotes depending on the degrees of f(x) and g(x).

Recommended video:

6:04

Intro to Rational Functions

Related Practice

Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0⁺ 1 / 3x

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

Textbook Question

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 3 (−2 / (x − 3)²) = −∞

Textbook Question

The sign-preserving property of continuous functions Let f be defined on an interval (a, b) and suppose that f(c) ≠ 0 at some c where f is continuous. Show that there is an interval (c − δ, c + δ) about c where f has the same sign as f(c).

Textbook Question

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x → ∞ √((8x² − 3) / (2x² + x))

Textbook Question

Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer.