Problem 113 - A Graphical Approach to Precalculus with Limits

Question

\(f(x)=\frac{x^{5}+x^{4}+x^{2}+1}{x^{4}+1}\) becomes \(f(x)=x+1+\frac{x^{2}-x}{x^{4}+1}\) after the numerator is divided by the denominator. (a) What is the equation of the oblique asymptote of the graph of the function? (b) For what \(x\) -value(s) does the graph of the function intersect its asymptote? (c) As \(x \rightarrow \infty,\) does the graph of the function approach its asymptote from above or below?

Step-by-Step Solution

Answer

(a) \( y = x + 1 \); (b) \( x = 0 \) and \( x = 1 \); (c) from above.

Step 1: Identify the Asymptote Equation

The given expression for the function is \( f(x) = x + 1 + \frac{x^2 - x}{x^4 + 1} \). The integer part, \( x + 1 \), represents the oblique asymptote of the function. Therefore, the equation of the oblique asymptote is \( y = x + 1 \).

Step 2: Find Intersection Points

To find where the function intersects the oblique asymptote, set \( f(x) = x + 1 \).So, \( x + 1 + \frac{x^2 - x}{x^4 + 1} = x + 1 \). Thus, \( \frac{x^2 - x}{x^4 + 1} = 0 \).The fraction \( \frac{x^2 - x}{x^4 + 1} = 0 \) only when the numerator \( x^2 - x = 0 \). Hence, \( x(x - 1) = 0 \), giving solutions \( x = 0 \) and \( x = 1 \).Therefore, the function intersects its asymptote at \( x = 0 \) and \( x = 1 \).

Step 3: Determine Behavior Approaching Infinity

Examine the behavior of \( \frac{x^2 - x}{x^4 + 1} \) as \( x \to \infty \).Notice \( \frac{x^2 - x}{x^4 + 1} = \frac{x(x-1)}{x^4 + 1} \), where as \( x \to \infty \), the dominant term in the denominator \( x^4 \) grows faster than any term in the numerator \( x^2 - x \), leading to the fraction approaching 0.Thus, \( f(x) = x + 1 + \frac{x^2 - x}{x^4 + 1} \to x + 1 + 0 = x + 1 \) from above as \( x \to \infty \).

Key Concepts

Polynomial Division

Polynomial division is a mathematical process used to divide one polynomial by another, resulting in a quotient and possibly a remainder. It's quite similar to the long division we use with numbers, but tailored for algebraic expressions.

When dealing with functions that involve polynomials, such as in our original exercise, polynomial division helps us to simplify the expression. We separate the rational function into a polynomial part and a fractional remainder. For example, dividing the numerator \(x^5 + x^4 + x^2 + 1\) by the denominator \(x^4 + 1\) yields \(x + 1\) with a remainder \(\frac{x^2 - x}{x^4 + 1}\).

This method is crucial for identifying characteristics of the function, like asymptotes, by breaking down complex polynomials into more manageable components. The outcome, \(x + 1\), is particularly significant here because it tells us the function's oblique asymptote, explaining how the graph behaves at large values of \(x\).

Remember, the goal of carrying out polynomial division is to simplify a function so that its long-term behavior becomes evident.

Intersection Points

To find where a function intersects its asymptote, you need to set the function equal to the asymptote and solve for \(x\). For our function \(f(x) = x + 1 + \frac{x^2 - x}{x^4 + 1}\), the asymptote is \(y = x + 1\).

The intersection occurs where:

• \(f(x) = x + 1\)
• Simplifies to: \(x + 1 + \frac{x^2 - x}{x^4 + 1} = x + 1\)

By solving \(\frac{x^2 - x}{x^4 + 1} = 0\), we find where the numerator is zero since the denominator can never equal zero unless specified otherwise.

The expression \(x^2 - x = 0\) factors to \(x(x - 1) = 0\). Hence, we have two solutions, \(x = 0\) and \(x = 1\).

Therefore, the graph intersects its asymptote precisely at these points \(x = 0\) and \(x = 1\). This means that for these \(x\)-values, the height of the function exactly matches the line of the oblique asymptote.

Behavior at Infinity

Understanding the behavior of a function as \(x\) goes to infinity, or examining its end behavior, gives insight into how the graph approaches an asymptote.

For the function \(f(x) = x + 1 + \frac{x^2 - x}{x^4 + 1}\), the key component to consider is the fraction \(\frac{x^2 - x}{x^4 + 1}\).

As \(x\) becomes very large (approaches infinity), the term \(x^4\) in the denominator grows much faster than \(x^2 - x\) in the numerator. This implies that the fraction itself shrinks towards zero:

• The larger the \(x\), the smaller the impact of this term on \(f(x)\).
• The function essentially becomes \(f(x) \approx x + 1\) as \(x\) increases.

As a result, the graph approaches its asymptote \(y = x + 1\) from above. This means though the curve gets infinitely close to the line, it trails from a higher position before converging completely.