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Chapter 2: Problem 38

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$f(x)=\frac{15 x}{3 x^{2}+1}$$

Short Answer

Expert verified
The horizontal asymptote of the function \(f(x)=\frac{15x}{3x^{2}+1}\) is \(y=0\).

Step by step solution

01

Identify the order of the numerator and the denominator

The order (or degree) of the numerator of the given function \(f(x)=\frac{15x}{3x^{2}+1}\) is 1 and the order of the denominator is 2. When the order of the denominator is greater than the numerator, the horizontal asymptote is at \(y=0\)
02

Determine the limit as x approaches infinity

To determine the limit as x approaches infinity, one needs to divide the numerator and the denominator by \(x^{2}\) (since 2 is the highest order of the denominator or numerator). After that, we determine the limit: \[ \lim_{x\to\infty} \frac{(15/x)}{(3+1/x^{2})}\]Since \(15/x\) and \(1/x^{2}\) tends to 0 when x tends to infinity, the limit of the function as x approaches infinity is zero.
03

Identify the horizontal asymptote

The value of the function as x approaches infinity is zero, so the line \(y=0\) is the horizontal asymptote of the function.

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Key Concepts

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

Rational Functions
A rational function is a type of function that can be defined as the ratio of two polynomials. This means it can be expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). Understanding rational functions involves recognizing both their algebraic structure and how they behave graphically.
  • They are characterized by polynomial expressions in both the numerator \( P(x) \) and the denominator \( Q(x) \).
  • The domain of these functions is restricted by the zeros of the denominator, as division by zero is undefined.
  • Common features of rational functions include vertical asymptotes, horizontal asymptotes, and removable discontinuities.
To find horizontal asymptotes, particularly, we look at the behavior of the function as \( x \) approaches positive or negative infinity. This involves checking the relative degrees of the numerator and the denominator, which brings us to our next concept: degrees of polynomials.
Degrees of Polynomials
The degree of a polynomial is a critical concept in understanding the behavior of polynomials in rational functions. It refers to the highest power of \( x \) in a polynomial and gives us insight into the end behavior of the function.
  • The degree helps in determining the leading term, which significantly influences the graph's shape as \( x \) moves towards infinity.
  • For the polynomial \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \), the degree is \( n \).
When comparing polynomials in a rational function, the degree of both the numerator and the denominator is assessed:
  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
  • If both have the same degree, the horizontal asymptote is given by the ratio of the leading coefficients.
  • If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote.
Knowing the degree helps in calculating limits at infinity, as shown in the next section.
Limits at Infinity
Limits help in understanding the behavior of a function as the input \( x \) becomes very large (positively or negatively). In rational functions, limits at infinity play a crucial role in finding horizontal asymptotes.
  • The focus is on determining \( \lim_{x\to\infty} f(x) \), which is critical when looking for horizontal asymptotes.
  • Evaluating limits at infinity often involves simplifying the rational function by dividing each term by \( x^n \), where \( n \) is the highest degree present in either the numerator or denominator.
  • Ultimately, these limits indicate the horizontal line that the graph of the function approaches but never actually reaches or crosses.
In the given function \( f(x)=\frac{15x}{3x^{2}+1} \), dividing each term by \( x^2 \) simplifies the function and reveals that as \( x \to \infty \), terms like \( 15/x \) and \( 1/x^2 \) approach zero. Thus, the horizontal asymptote is the line \( y = 0 \). This result aligns with both the degree analysis and the behavior of rational functions at infinity.

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Most popular questions from this chapter

Write the equation of each parabola in standard form. Vertex: \((-3,-1) ;\) The graph passes through the point \((-2,-3)\)

The heat generated by a stove element varies directly as the square of the voltage and inversely as the resistance. If the voltage remains constant, what needs to be done to triple the amount of heat generated?

Will help you prepare for the material covered in the next section. Use $$\frac{2 x^{3}-3 x^{2}-11 x+6}{x-3}=2 x^{2}+3 x-2$$ to factor \(2 x^{3}-3 x^{2}-11 x+6\) completely.

Solve and graph the solution set on a number line: $$\frac{2 x-3}{4} \geq \frac{3 x}{4}+\frac{1}{2}$$ (Section P.9, Example 5 )

Explain how to decide whether a parabola opens upward or downward.
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