Problem 17 - Calculus Early Transcendentals

Question

Show that \(y=x+3\) is an oblique asymptote of the graph of \(f(x)=x^{2} /(x-3) .\) Sketch the graph of \(y=f(x)\) showing this asymptotic behavior.

Step-by-Step Solution

Answer

The oblique asymptote is \(y = x + 3\).

Step 1: Definition of Asymptote

An oblique asymptote is a linear function that the graph of a given function approaches as the independent variable either increases or decreases without bound. For the function \( f(x) = \frac{x^2}{x-3} \), we need to find a line \( y = mx + c \) that satisfies \( \lim_{{x \to \infty}} (f(x) - (mx + c)) = 0 \).

Step 2: Long Division of Polynomial

To find the oblique asymptote, perform polynomial long division on \( \frac{x^2}{x-3} \). Divide \( x^2 \) by \( x-3 \). The first term is \( x \), which when multiplied back gives \( x^2 - 3x \). Subtracting, we have \( 3x \). Next, divide \( 3x \) by \( x-3 \) to get \( 3 \), which, when multiplied back, gives \( 3x - 9 \). Subtract to get a remainder of \( 9 \). So we have \( \frac{x^2}{x-3} = x + 3 + \frac{9}{x-3} \).

Step 3: Identify the Oblique Asymptote

The result from the division shows that \( f(x) = x + 3 + \frac{9}{x-3} \). As \( x \to \infty \), the term \( \frac{9}{x-3} \) approaches \( 0 \), so the function approaches \( y = x + 3 \). Thus, the oblique asymptote is \( y = x + 3 \).

Step 4: Sketch the Graph with Asymptote

To sketch the graph, note that this function has a vertical asymptote at \( x = 3 \), where the denominator is zero. Draw the oblique asymptote \( y = x + 3 \) as a dotted line. The graph of \( f(x) \) will approach this line as \( x \to \infty \) and \( x \to -\infty \). Make sure the graph reflects this behavior, crossing through the origin when \( x = 0 \) since \( f(0) = 0 \).

Key Concepts

Polynomial Long Division

To find the oblique asymptote of a function like \( f(x) = \frac{x^2}{x-3} \), polynomial long division is the technique you need. This approach helps you break down a rational function into simpler parts, making it easier to see the function's behavior at the extremes.
In this context, we divide \(x^2\) by \(x-3\). Start by recognizing that the highest degree term in the numerator, \(x^2\), is divided by the highest degree term in the denominator, \(x\). The result of this division gives us \(x\).

• Multiply \(x\) by \(x-3\) to get \(x^2 - 3x\).
• Subtract \(x^2 - 3x\) from the original \(x^2\), which leaves \(3x\).
• Then, divide \(3x\) by \(x-3\), which simplifies down to a result of \(3\).

The next step is to multiply and subtract this result again, giving a remainder of 9. Therefore, the division gives us \(\frac{x^2}{x-3} = x + 3 + \frac{9}{x-3}\).
This equation now easily reveals the behavior as \(x\) becomes extremely large or small.

Vertical Asymptote

Vertical asymptotes occur where a rational function's denominator becomes zero, resulting in the expression being undefined. For the function \( f(x) = \frac{x^2}{x-3} \), we identify a vertical asymptote at \( x = 3 \). This is because substituting \( x = 3 \) makes the denominator zero, leading to a division by zero scenario, which is undefined.
Graphically, vertical asymptotes are depicted as dashed vertical lines. They indicate where the function grows indefinitely in either the positive or negative direction close to that specific \(x\) value. They act as a boundary that the curve of the function will approach but never cross or touch. When sketching the graph, this asymptote helps define regions of behavior, showing how the curve behaves as it nears \(x = 3\).
Understanding vertical asymptotes is crucial, as they help to predict the "unbounded" behavior of a graph when near these points.

Asymptotic Behavior

Asymptotic behavior showcases how a function behaves as the input value, \(x\), grows very large or very small (approaches infinity or negative infinity). For \( f(x) = \frac{x^2}{x-3} \), the function has an oblique asymptote, \( y = x + 3 \). Compared to horizontal asymptotes which are horizontal lines, an oblique asymptote is slanted and arises when the degree of the polynomial in the numerator is one more than in the denominator.
Having simplified using polynomial long division, we get \( f(x) = x + 3 + \frac{9}{x-3} \). As \( x \to \infty \), the term \( \frac{9}{x-3} \) tends toward zero, meaning \( f(x) \) closely aligns with \( y = x + 3 \).

• The function's curve gets closer to this line but without meeting it exactly.
• In essence, it tells us the direction and angle the graph will trend toward as \(x\) becomes very large or small.

Grasping asymptotic behavior allows us to approximate and predict the graph's trajectory over long intervals, aiding in creating a more comprehensive sketch of the function's behavior.