Question

Determine \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of \(f\) (if any). $$f(x)=\frac{3 x^{2}-7}{x^{2}+5 x}$$

Short answer

Question: Determine the horizontal asymptote of the function \(f(x) = \frac{3x^2-7}{x^2+5x}\). Answer: The horizontal asymptote of the function \(f(x) = \frac{3x^2-7}{x^2+5x}\) is at \(y = 3\).

Step-by-step solution

Identify the degree of the numerator and denominator

Let's first identify the degree of the numerator and the denominator of the rational function. The degree of a term is the highest exponent of its variable. Numerator: \(3 x^2 - 7\) has a degree of 2 Denominator: \(x^2 + 5x\) also has a degree of 2.

Determine the coefficients of the highest power terms

Now we identify the leading coefficients of the numerator and the denominator, which are the coefficients of the highest power terms in each. Numerator: The coefficient of the term \(3x^2\) is \(3\). Denominator: The coefficient of the term \(x^2\) is \(1\).

Evaluate the limits as x approaches infinity and negative infinity

With the information from Steps 1 and 2, we can use the rule that if the degrees of the numerator and denominator are equal, then the limit as x approaches infinity or negative infinity of the function is equal to the ratio of the leading coefficients. So, \(\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{3x^2-7}{x^2+5x} = \frac{3}{1} = 3\) Similarly, \(\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} \frac{3x^2-7}{x^2+5x} = \frac{3}{1} = 3\)

Determine the horizontal asymptote of the function

Since both the limits as \(x\) approaches infinity and negative infinity are equal to 3, we have a horizontal asymptote at the line \(y = 3\). So, the function \(f(x) = \frac{3x^2-7}{x^2+5x}\) has a horizontal asymptote at \(y = 3\).

Key concepts

Horizontal Asymptote

In the context of rational functions, a horizontal asymptote is a straight horizontal line that the graph of the function approaches as the value of the independent variable becomes extremely large or extremely small. It helps us understand the end behavior of a function.
For the function in this exercise, we found that the horizontal asymptote is at the line where \( y = 3 \). This means that as \( x \) heads towards infinity, both positive and negative, the value of \( f(x) \) approaches 3.

• If the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients of the highest power terms.
• Check if the function approaches a constant value as \( x \to \infty \) or \( x \to -\infty \). If it does, that constant is the horizontal asymptote.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials. Typically, it is written as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
In the given exercise, \( f(x) = \frac{3x^2 - 7}{x^2 + 5x} \) is a rational function. Here, the numerator is \( 3x^2 - 7 \) and the denominator is \( x^2 + 5x \).

• Rational functions can have vertical asymptotes and horizontal asymptotes. Vertical asymptotes occur where the function becomes undefined, typically where the denominator is zero.
• They can also create interesting curves and end behaviors, dependent on the degree and coefficients of the polynomials. Understanding rational functions is key to mastering calculus and algebra.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial expression. It is crucial in determining the behavior and limits of functions, such as rational functions.
In the exercise provided, both the numerator, \( 3x^2 - 7 \), and the denominator, \( x^2 + 5x \), have a degree of 2. This equality is why the horizontal asymptote occurs at the ratio of their leading coefficients, which is 3.

• The degree influences the end behavior of a polynomial function. For example, a parabola (quadratic function) has a degree of 2, resulting in a "U"-shaped curve.
• In rational functions, the relationship between the degrees of numerator and denominator dictates the horizontal asymptote. Equal degrees suggest a horizontal asymptote at the ratio of their leading coefficients.