Question

The line \(y=a x+b\) is called an oblique asymptote to the graph of \(y=f(x)\) if either \(\lim _{x \rightarrow \infty}[f(x)-(a x+b)]=0\) or \(\lim _{x \rightarrow-\infty}[f(x)-(a x+b)]=0 .\) Find the oblique asymptote for $$ f(x)=\frac{2 x^{4}+3 x^{3}-2 x-4}{x^{3}-1} $$

Short answer

The oblique asymptote is \( y = 2x + 5 \).

Step-by-step solution

Analyze the Degree of the Polynomial

First, we should compare the degrees of the polynomial in the numerator and the denominator of the function \( f(x) = \frac{2x^4 + 3x^3 - 2x - 4}{x^3 - 1} \). The numerator has a degree of 4 and the denominator has a degree of 3.

Perform Polynomial Long Division

Since the degree of the numerator is higher than the degree of the denominator, we perform polynomial long division on \( \frac{2x^4 + 3x^3 - 2x - 4}{x^3 - 1} \). Divide \(2x^4 \) by \(x^3\) to get \(2x\). Multiply \((x^3 - 1)\) by \(2x\) and subtract the result from the original numerator.

Simplify the Expression

After performing the division described in Step 2, the result simplifies to \(2x + 5 + \frac{-2x - 4}{x^3 - 1}\). Here, \(2x + 5\) is the quotient and represents the linear part we are looking for.

Determine the Oblique Asymptote

The expression for the function can be written as the quotient \(2x + 5\), plus the remainder \( \frac{-2x - 4}{x^3 - 1} \). As \(x \to \infty\), the remainder part approaches 0. Therefore, the oblique asymptote of \( f(x) \) is \( y = 2x + 5 \).

Key concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to how you divide numbers. It becomes especially useful when you encounter a fraction of polynomials, like in our original exercise, where the numerator's polynomial degree is greater than the denominator's.

Here's how it works:

• Start by dividing the first term of the numerator by the first term of the denominator.
• Multiply the entire divisor by this result and subtract it from the original numerator polynomial. This provides you with a new polynomial.
• Repeat the process using the new polynomial as the numerator.

Continue this process until the degree of the new numerator polynomial is less than the degree of the denominator. The result will be a quotient and a remainder. In our case, when dividing \(2x^4 + 3x^3 - 2x - 4\) by \(x^3 - 1\), we found that the quotient is \(2x + 5\) and the remainder is \(-2x - 4\). This division helps us identify the oblique asymptote of the function.

Limit at Infinity

The limit of a function as \(x\) approaches infinity helps determine the function's behavior as the variable grows larger. Specifically, for asymptotes, we're interested in whether the remainder of a polynomial fraction approaches zero.

In terms of oblique asymptotes, after performing polynomial long division, we express the function as a combination of a quotient and a remainder. The quotient is a linear function, and the remainder is a fraction that approaches zero as \(x\) becomes very large. Thus, if \(\lim_{x \to \infty} [f(x) - (ax + b)] = 0\), it means that\(f(x)\) closely follows the line \(y = ax + b\), as \(x\) increases. In our solution, the limit of the remainder part of the expression approaches zero, confirming \(y = 2x + 5\) as the oblique asymptote.

Degree of Polynomials

Understanding the degree of a polynomial is crucial in determining an asymptote. The degree is simply the highest power of \(x\) in a polynomial expression. For example, in the polynomial \(2x^4 + 3x^3 - 2x - 4\), the degree is 4 because \(x^4\) is the term with the highest power.

When dealing with fractions of polynomials, comparing the degree of the numerator and the denominator helps predict the type of asymptote. If the numerator's degree is greater than the denominator's, the result is typically an oblique or slant asymptote. This is because the polynomial long division will yield a linear result (quotient). In our case, having numerator degree 4 and denominator degree 3 indicates the presence of an oblique asymptote.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a certain value, typically as \(x\) goes to infinity. Such behavior reveals that the function approaches a line (the asymptote), but never actually reaches or touches it.

In the context of oblique asymptotes, understanding asymptotic behavior helps us realize that, while the function \(f(x)\) gets closer to linear \(y = ax + b\) at extremes, the two will never intersect precisely. This behavior is crucial for graphing and understanding functions, predicting their trend when extended too large values of \(x\).

Furthermore, testing moments illustrates this tendency, showing the convergence of \(f(x) - (ax + b)\) to zero as \(x\) approaches infinity or negative infinity. In brief, not only does the function approach or moves along an asymptote, but it does so without actually hitting it, providing a reliable pattern for anticipating the behavior of complex functions.