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Long division is a method used to divide polynomials, much like how you would divide numbers. It's a systematic approach that breaks down complex polynomials into simpler components. When solving for a slant asymptote, long division helps isolate the leading term that corresponds to the slant asymptote.

• Start with the dividend, which is the polynomial in the numerator. For this function, it's the polynomial \(x^4 - 1\).
• The divisor is the polynomial in the denominator, which for this example is \(x^3 + x\).
• The division process continues by dividing the leading term of the dividend by the leading term of the divisor.
• This will give you the first term of the quotient, which in our example is \(x\).

As you complete each step of the division, you subtract the product of the divisor and the current term of the quotient from the current dividend. Continue this process, bringing down terms as needed, until no terms remain that can be divided further. The polynomial we extract from this process helps us simplify the function to identify its asymptotic behavior.

Polynomial division works hand-in-hand with long division but focuses more broadly on dividing one polynomial by another, resulting in a quotient and a remainder. Unlike simple long division used in arithmetic, polynomial division requires careful handling of variables and their coefficients.Through polynomial division, we start by simplifying the function, which makes it easier to analyze the behavior at infinity. For the function \(f(x) = \frac{x^4 - 1}{x^3 + x}\), the division yields a quotient of \(x\) with a remainder of \(-\frac{1}{x^2} + \frac{1}{x^4}\).

• The quotient represents the main term responsible for defining the slant of the asymptote.
• The remainder, which comprises terms that diminish as \(x\) becomes very large, becomes negligible in determining the asymptote.

This division, by simplifying the relation between the numerator and the denominator, enables clearer glimpses into how the function behaves as \(x\) approaches the extremes.

Graphing functions provides a visual representation of how functions behave, making it easier to comprehend the relationship between the function and its slant asymptote. For \(f(x) = \frac{x^4 - 1}{x^3 + x}\), graphing allows us to visibly confirm the behavior as \(x\) approaches infinite values.

• First, plot the function \(f(x)\) using a graphing calculator or software. This enables you to get a basic sketch of how the curve behaves.
• Next, plot the slant asymptote \(y = x\), which is a straight line with a slope of 1 passing through the origin.
• Observe how the graph of \(f(x)\) gets closer and closer to this line as \(x\) moves towards positive or negative infinity.

This overlap with the slant asymptote showcases how the function aligns along the line beyond finite values. It allows better insight into both the vertical and horizontal behaviors that occur at the limits.

Understanding limits at infinity is crucial for identifying slant asymptotes. Limits at infinity help describe the behavior of a function as the input becomes arbitrarily large in the positive or negative direction.For the slant asymptote to exist for \(f(x)\), the expression \(f(x) - (mx + b)\) must approach 0 as \(x\) approaches infinity. This is examined mathematically by evaluating limits such as\[ \lim_{x \to \infty} \left( f(x) - (mx + b) \right) = 0 \]In our example, as \(x\) moves towards infinity,

• The remainder \(-\frac{1}{x^2} + \frac{1}{x^4}\) approaches 0 rapidly as these terms diminish.
• This leaves the dominant term \(x\), which aligns with the slant asymptote given by \(y = x\).

Examining both \(x\) to the positive infinity and negative infinity ensures symmetry in understanding the function's behavior. Therefore, limits at infinity are pivotal in affirming the existence and direction of an asymptote.