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Chapter 2: Problem 18

Determine the horizontal asymptote of each function. If none exists, state that fact. $$ f(x)=\frac{8 x^{4}-5 x^{2}}{2 x^{3}+x^{2}} $$

Short Answer

Expert verified
The function does not have a horizontal asymptote.

Step by step solution

01

Understand the Problem

We need to determine the horizontal asymptote of the function \( f(x) = \frac{8x^4 - 5x^2}{2x^3 + x^2} \). To do this, we must examine the behavior of the function as \( x \) approaches infinity.
02

Identify the Polynomial Degrees

Identify the degrees of the numerator and the denominator. The degree of the numerator \( 8x^4 - 5x^2 \) is 4, and the degree of the denominator \( 2x^3 + x^2 \) is 3.
03

Compare the Degrees of the Polynomials

Compare the degrees of the numerator and the denominator. The numerator's degree (4) is greater than that of the denominator (3).
04

Determine the Horizontal Asymptote

For rational functions, if the degree of the numerator is greater than the degree of the denominator, the function does not have a horizontal asymptote. Instead, the function has an oblique (slant) asymptote, which is beyond the scope of this question about horizontal asymptotes.

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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 fractions where both the numerator and the denominator are polynomial expressions. For example, consider the function \( f(x) = \frac{8x^4 - 5x^2}{2x^3 + x^2} \). In this function, the numerator is \( 8x^4 - 5x^2 \) and the denominator is \( 2x^3 + x^2 \). Both are polynomials because they are composed of terms formed by multiplying constants and non-negative integer powers of the variable \(x\).

When dealing with rational functions, it is crucial to consider the behavior of these functions as \( x \) approaches -鈭� or +鈭�. This is where concepts such as vertical and horizontal asymptotes come into play.
  • Vertical asymptotes occur where the function is undefined, often related to the denominator being zero.
  • Horizontal asymptotes provide information on the behavior of the function as it extends towards infinity or negative infinity.
Exploring Polynomial Degrees
The degree of a polynomial is the highest power of the variable in the polynomial expression. In a rational function, each polynomial (numerator and denominator) has its respective degree, which is vital in understanding the function's long-term behavior.
  • For the numerator \(8x^4 - 5x^2\), the degree is 4 because the highest power of \(x\) is \(x^4\).
  • For the denominator \(2x^3 + x^2\), the degree is 3, due to the highest power being \(x^3\).
The degrees of these polynomials help determine how they compare to each other. This comparison is essential in deciding the presence and nature of any horizontal asymptotes in the rational function.

Understanding polynomial degrees also assists in anticipating the end behavior of functions, providing insights into how the graphs behave as \(x\) becomes extremely large or small.
Comparing Asymptotes Through Polynomial Degrees
To determine if a horizontal asymptote exists in a rational function, it is essential to compare the degrees of the numerator and the denominator.
  • If the numerator's degree is larger than the denominator's, as in \( f(x) = \frac{8x^4 - 5x^2}{2x^3 + x^2} \), the function has no horizontal asymptote, although it may have a slant asymptote.
  • If the numerator's degree is less than the denominator's, there is a horizontal asymptote at \( y = 0 \).
  • If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
In our example, the presence of a higher degree in the numerator demonstrates that as \( x \) becomes very large or small, the function will not level off to a constant value. This behavior points to a lack of a horizontal asymptote, guiding us to expect possibly different behavior, such as forming a slant asymptote.

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Most popular questions from this chapter

A power line is to be constructed from a power station at point \(A\) to an island at point \(C,\) which is \(l\) mi directly out in the water from \(a\) point \(B\) on the shore. Point \(B\) is 4 mi downshore from the power station at \(A\). It costs \(\$ 5000\) per mile to lay the power line under water and \(\$ 3000\) per mile to lay the line under ground. At what point \(S\) downshore from \(A\) should the line come to the shore in order to minimize cost? Note that \(S\) could very well be \(B\) or \(A\). (Hint: The length of \(C S\) is \(\left.\sqrt{1+x^{2}} .\right)\)

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$f(x)=(x-1)^{3}$$

Find the absolute extrema of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real numbers, \((-\infty, \infty)\). $$g(x)=\frac{1}{3} x^{3}+2 x^{2}+x ; \quad[-4,0]$$

An inner city revitalization zone is a rectangle that is twice as long as it is wide. A diagonal through the region is growing at a rate of \(90 \mathrm{~m}\) per year at a time when the region is \(440 \mathrm{~m}\) wide. How fast is the area changing at that point in time?

Graph each of the following equations. Equations must be solved for \(y\) before they can be entered into most calculators. Graphicus does not require that equations be solved for \(y .\) $$ y^{2}=x^{3} $$
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