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Chapter 1: Problem 51

In Problems \(50-51\), explain what is wrong with the statement. Every rational function has a horizontal asymptote.

Short Answer

Expert verified
Not all rational functions have horizontal asymptotes; it depends on the degrees of the numerator and denominator.

Step by step solution

01

Understanding Rational Functions

A rational function is a quotient of two polynomials. It is expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). This definition is crucial as it helps us explore the behavior of these functions, especially as \( x \) approaches infinity or negative infinity.
02

Define Horizontal Asymptote

A horizontal asymptote of a function \( f(x) \) represents a horizontal line \( y = c \) that the graph of \( f \) approaches as \( x \) tends toward infinity or negative infinity. For rational functions, horizontal asymptotes depend on the degrees of the numerator and denominator polynomials.
03

Determine Conditions for Horizontal Asymptotes in Rational Functions

For a rational function \( \frac{P(x)}{Q(x)} \), we can determine horizontal asymptotes by comparing the degrees of \( P(x) \) and \( Q(x) \): 1. If the degree of the numerator is less than the degree of the denominator, \( y = 0 \) is a horizontal asymptote. 2. If the degrees are equal, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of \( P(x) \) and \( Q(x) \), respectively. 3. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote; instead, there may be an oblique asymptote.
04

Analyze the Statement

The statement "Every rational function has a horizontal asymptote" is incorrect because there exist rational functions with numerator degrees greater than their denominators. In such cases, the function does not have a horizontal asymptote, which contradicts the given statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Rational Functions
Rational functions are at the heart of understanding many advanced mathematical concepts. These functions are formed by dividing two polynomials. For any rational function, you can express it as \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. It's crucial to remember that the polynomial \( Q(x) \) cannot be zero. This form gives us insight into the behavior of the function, particularly as \( x \) grows very large or very small.
  • The numerator \( P(x) \) is a polynomial representing the upper part of the rational expression.
  • The denominator \( Q(x) \) ensures the function's value is well-defined, apart from points where \( Q(x) = 0 \), which can cause the function to be undefined or create vertical asymptotes.
Recognizing the structure of rational functions helps when analyzing how they behave at extreme values, which directly relates to the identification of asymptotes.
Horizontal Asymptotes in Functions
A horizontal asymptote is a horizontal line \( y = c \) that a graph approaches as \( x \) moves towards infinity or negative infinity. For many students, this can initially be a confusing concept, but understanding it involves observing how a function "flattens out" or trends close to a specific line.
  • Approach at Infinity: As \( x \) gets larger and larger, the function values might become closer to a specific constant value.
  • Two-Way Approach: It's important to check the behavior as \( x \) approaches both positive and negative infinity.
In the context of rational functions, whether a horizontal asymptote exists depends largely upon comparing the degrees of the polynomials in the numerator and denominator. The horizontal line that a graph may approach provides valuable insight into the long-term behavior of the function.
Polynomial Degree Comparison in Rational Functions
Understanding the relationship between the degrees of the polynomials in a rational function is key to predicting its asymptotic behavior. This comparison is essential for determining if and where horizontal asymptotes appear.Here are the rules:
  • Numerator degree less than denominator: The horizontal asymptote is at \( y = 0 \). This occurs when the polynomial in the numerator grows slower compared to the denominator as \( x \) increases.
  • Degrees are equal: The asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are leading coefficients of the numerator and denominator respectively. This means they grow at the same rate and stabilize into a ratio of these leading terms.
  • Numerator degree greater than denominator: No horizontal asymptote exists. Instead, another type of asymptote, such as an oblique or non-linear asymptote, may occur since the numerator eventually dominates the behavior of the function.
By understanding these degree conditions, students can quickly evaluate the presence and position of horizontal asymptotes in rational functions.

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Most popular questions from this chapter

Are the statements true or false? Give an explanation for your answer. If a function is odd, then it does not have an inverse.

Suppose that \(\lim _{x \rightarrow 3} f(x)=7 .\) Are the statements in Problems \(89-95\) true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample. If \(\lim _{x \rightarrow 3} g(x)\) does not exist, then \(\lim _{x \rightarrow 3}(f(x) g(x))\) does not exist.

For the functions in Problems \(46-53,\) do the following: (a) Make a table of values of \(f(x)\) for \(x=0.1,0.01,0.001\) \(0.0001,-0.1,-0.01,-0.001,\) and -0.0001 (b) Make a conjecture about the value of \(\lim _{x \rightarrow 0} f(x)\) (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for \(x\) near 0 such that the difference between your conjectured limit and the value of the function is less than \(0.01 .\) (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.) $$f(x)=\frac{e^{x}-1}{x}$$

Determine functions \(f\) and \(g\) such that \(h(x)=f(g(x)) .\) [Note: There is more than one correct answer. Do not choose \(f(x)=x \text { or } g(x)=x\).] $$h(x)=(x+1)^{3}$$

For the functions in Problems \(46-53,\) do the following: (a) Make a table of values of \(f(x)\) for \(x=0.1,0.01,0.001\) \(0.0001,-0.1,-0.01,-0.001,\) and -0.0001 (b) Make a conjecture about the value of \(\lim _{x \rightarrow 0} f(x)\) (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for \(x\) near 0 such that the difference between your conjectured limit and the value of the function is less than \(0.01 .\) (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.) $$f(x)=\frac{\sin 3 x}{x}$$
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