Problem 34 - Calculus Early Transcendentals

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). $$f(x)=\frac{-x^{3}+1}{2 x+8}$$

Step-by-Step Solution

Answer

Answer: The limits of the function as x approaches positive and negative infinity are \(\lim _{x \rightarrow \infty} f(x) = -\infty\) and \(\lim _{x \rightarrow -\infty} f(x) = \infty\). There is no horizontal asymptote for the given function.

Step 1: Analyze the degree of the numerator and the degree of the denominator

In the given function \(f(x) = \frac{-x^3 + 1}{2x + 8}\), the degree of the numerator is 3, as the highest exponent of x in the numerator is 3. The degree of the denominator is 1, as the highest exponent of x in the denominator is 1.

Step 2: Determine the ratio of the degrees

Since the degree of the numerator is 3 and the degree of the denominator is 1, their ratio is \(\frac{3}{1}\), which is greater than 1.

Step 3: Find the limit as x approaches positive infinity

When the ratio of the degrees of the numerator and the denominator is greater than 1, the limit of the function as x approaches positive infinity will not exist, because the function will either go to positive or negative infinity depending on the leading coefficients. In our case, the leading coefficient of the numerator is -1, so as x approaches positive infinity, the function will go to negative infinity. Therefore, \(\lim _{x \rightarrow \infty} f(x) = -\infty\).

Step 4: Find the limit as x approaches negative infinity

Similarly, when the ratio of the degrees of the numerator and the denominator is greater than 1, the limit of the function as x approaches negative infinity will not exist. In our case, as x approaches negative infinity, the leading coefficient of the numerator -1 multiplied by a negative value will result in a positive value. Therefore, \(\lim _{x \rightarrow -\infty} f(x) = \infty\).

Step 5: Determine the horizontal asymptote

Since the limits as x approaches positive and negative infinity do not exist, there is no horizontal asymptote for the function \(f(x) = \frac{-x^3 + 1}{2x + 8}\). In conclusion, we have found that \(\lim _{x \rightarrow \infty} f(x) = -\infty\), \(\lim _{x \rightarrow -\infty} f(x) = \infty\), and there is no horizontal asymptote for the given function.

Key Concepts

Rational Functions

Rational functions are incredibly important in mathematics. They're essentially a ratio between two polynomial expressions. You can think of them as a fraction where the numerator and the denominator are polynomials. So, for the function \(f(x) = \frac{-x^3 + 1}{2x + 8}\), \(-x^3 + 1\) is the numerator, and \(2x + 8\) is the denominator.

What makes rational functions interesting is how they behave at extreme values of \(x\), such as when \(x\) approaches infinity or negative infinity. Their behavior is often determined by the degrees of these polynomials, which we will discuss more in the next section.

When analyzing rational functions, it's crucial to see how the growth rates of the numerator and denominator affect the overall function as \(x\) becomes very large or very small. Different ratios of these degrees lead to markedly different long-term behaviors of the function.

Degrees of Polynomials

In mathematics, the degree of a polynomial is the highest power of the variable \(x\) that appears in the polynomial equation. For example, in the function \(f(x) = \frac{-x^3 + 1}{2x + 8}\), the numerator \(-x^3 + 1\) has a highest power of 3 (making its degree 3), and the denominator \(2x + 8\) has a highest power of 1 (making its degree 1).

The degree of these polynomials is crucial when evaluating limits at infinity and determining asymptotic behavior. Specifically:

• If the degree of the numerator is greater than the degree of the denominator, as \(x\) approaches infinity or negative infinity, the function tends toward infinity or negative infinity. This means no horizontal asymptote exists.
• If the degree of the numerator is less than the degree of the denominator, the function approaches zero as \(x\) becomes very large or very small.
• If the degree of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.

Understanding these rules helps predict the behavior of rational functions at their extremes, being key to solving problems involving limits and asymptotes.

Horizontal Asymptotes

Horizontal asymptotes help in understanding the end behavior of a rational function as \(x\) grows large in magnitude, either positively or negatively. They determine where the function stabilizes, or flattens out.

Horizontal asymptotes can be found using the degrees of the polynomials in the rational function:

• If the degree of the numerator is greater than the degree of the denominator, there is generally no horizontal asymptote, as seen in our example \(f(x) = \frac{-x^3 + 1}{2x + 8}\). Here, the numerator's degree is 3 and the denominator's is 1, leading to limits of \(\pm\infty\).
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). This happens because the denominator grows much faster than the numerator, ultimately flattening the function's graph towards zero.
• If the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients in the polynomials of the numerator and denominator.

Knowing these distinctions helps in graphing functions and predicting their behavior at extreme values of \(x\), allowing us to visualize and understand the function's long-term tendencies.