Problem 522 - Precalculus

Question

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote $$f(x)=\frac{2 x}{(x-3)^{2}}$$

Step-by-Step Solution

Answer

Vertical asymptote at \(x = 3\); horizontal asymptote at \(y = 0\).

Step 1: Identify Vertical Asymptote

To identify the vertical asymptote of the function \(f(x) = \frac{2x}{(x-3)^2}\), set the denominator equal to zero: \((x-3)^2 = 0\) which implies \(x = 3\). Hence, there is a vertical asymptote at \(x = 3\).

Step 2: Identify Horizontal Asymptote

Since the degree of the numerator is 1 and the degree of the denominator is 2, the horizontal asymptote is determined by the limit as \(x\) approaches infinity. Thus, \(\lim_{{x \to \pm \infty}} f(x) = 0\). The horizontal asymptote is \(y = 0\).

Step 3: Create a Table Near Vertical Asymptote

Select values of \(x\) approaching 3 from both sides, such as 2.9, 2.99, 3.01, and 3.1. Calculate \(f(x)\) for these values. Observe how the values of \(f(x)\) behave as \(x\) gets closer to 3.

Step 4: Create a Table Near Horizontal Asymptote

Choose large positive and negative values for \(x\), such as 10, 100, -10, and -100. Compute \(f(x)\) and observe the approach of \(f(x)\) values towards 0.

Step 5: Analyze Table Results

For the first table, observe that as \(x\) approaches 3 from the left, \(f(x)\) becomes large and negative, while from the right, \(f(x)\) becomes large and positive (or vice versa depending on computation values). In the second table, as \(x\) becomes very large or very small, \(f(x)\) approaches 0.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are lines that a function approaches but never actually reaches as the input value of the function becomes extremely large in magnitude, either positively or negatively. Essentially, they describe the end behavior of a function.
For the rational function given in the exercise, \( f(x) = \frac{2x}{(x-3)^2} \), the degree of the numerator is 1 and the degree of the denominator is 2.

• If the degree of the numerator is less than the degree of the denominator, as it is here, the horizontal asymptote is \(y = 0\).
• In this scenario, as \(x\) approaches both positive and negative infinity, the function \(f(x)\) trends towards 0.

Understanding horizontal asymptotes helps us comprehend how a function "levels out" as \(x\) increases or decreases without bounds.

Rational Functions

Rational functions are expressions that are the ratio of two polynomials. The general form is \( f(x) = \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) \(eq\) 0 to prevent division by zero.

• An essential aspect of rational functions is identifying their asymptotic behavior.
• They may have both vertical and horizontal (or oblique) asymptotes depending on the degrees of the polynomials involved.

The function \( f(x) = \frac{2x}{(x-3)^2} \) is a perfect example of a rational function with both vertical and horizontal asymptotes.
The degree of the numerator (1) is less than the degree of the denominator (2), providing a horizontal asymptote at \(y = 0\).

Limits in Calculus

In calculus, limits help us understand the behavior of functions as the input approaches a specific value or infinity.
The vertical asymptote occurs where the denominator equals zero, as seen with \( f(x) = \frac{2x}{(x-3)^2} \), at \(x = 3\). Being able to spot where a function might "explode" towards positive or negative infinity, allows using limits to better predict function behavior.
When discussing the horizontal asymptote at \(y = 0\), the following limit is essential:

• \( \lim_{{x \to \pm \infty}} \frac{2x}{(x-3)^2} = 0 \) describes how \(f(x)\) behaves as \(x\) moves far from the origin in either direction.

Limits are a cornerstone in calculus for exploring a function's tendencies near these unique points and infinities.

Function Behavior Near Asymptotes

Asymptotes provide insight into the behavior of functions close to particular lines or points.

• Vertical asymptotes, like the one at \(x = 3\) in \(f(x) = \frac{2x}{(x-3)^2}\), indicate where a function can increase or decrease without bound. Approaching \(x = 3\) from either side, you'll notice \(f(x)\) becomes indefinitely large or small.
• Horizontal asymptotes, such as \(y = 0\), depict how a function behaves as \(x\) trends towards positive or negative infinity. It hints at a settling behavior of the function's values.

Creating tables, as described in the solution, is a great way to observe and understand these behaviors. By selecting values near the asymptotes and calculating \(f(x)\), you can see these infinite or zeroing behaviors in numerical form.