Find the Asymptotes (x^2-x-6)/(x^2-6x+8)
The problem presented is a question of calculus or precalculus, specifically related to the study of rational functions. The task is to find the asymptotes of the given function, which is the ratio of two polynomials: (x² - x - 6) / (x² - 6x + 8). To find the asymptotes of a rational function, one typically looks for both vertical and horizontal (or oblique) asymptotes.
Vertical asymptotes occur where the denominator of the rational function is zero, provided that the numerator is not also zero at those points, because this suggests that the function will tend to infinity or negative infinity as it approaches those points from the left or the right.
Horizontal or oblique asymptotes are related to the behavior of the function at infinity. They describe the line that the function approaches as x goes to positive or negative infinity. The determination of horizontal asymptotes involves comparing the degrees of the numerator and denominator polynomials, while finding oblique asymptotes may involve polynomial long division or other techniques if the numerator is of a higher degree than the denominator.
The question requires the analysis of both the horizontal/oblique and vertical asymptotes of the given rational function.
$\frac{x^{2} - x - 6}{x^{2} - 6 x + 8}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the function $\frac{x^{2} - x - 6}{x^{2} - 6 x + 8}$ to be undefined. These are $x = 2$ and $x = 4$.
Step 2:
Observe the behavior of the function as $x$ approaches 2. It tends towards negative infinity from the left and positive infinity from the right, indicating a vertical asymptote at $x = 2$.
Step 3:
Similarly, as $x$ approaches 4, the function goes to negative infinity from the left and positive infinity from the right, confirming another vertical asymptote at $x = 4$.
Step 4:
Compile a list of all vertical asymptotes found: $x = 2$ and $x = 4$.
Step 5:
Examine the degrees of the numerator and denominator for a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$. The horizontal asymptote depends on the relationship between $n$ and $m$:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but possibly an oblique asymptote.
Step 6:
Determine the degrees of the numerator ($n$) and the denominator ($m$). For our function, both $n$ and $m$ are 2.
Step 7:
Since the degrees are equal ($n = m$), the horizontal asymptote is given by $y = \frac{a}{b}$. Here, $a = 1$ and $b = 1$, so the horizontal asymptote is $y = 1$.
Step 8:
There are no oblique asymptotes since the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize all asymptotes:
- Vertical Asymptotes: $x = 2$ and $x = 4$
- Horizontal Asymptote: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, we follow these steps:
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero, as long as the numerator is not also zero at those points. To find them, solve the equation set by the denominator equal to zero.
Behavior Near Vertical Asymptotes: To confirm that a vertical asymptote exists, check the limits of the function as $x$ approaches the value from the left and right. If the function approaches infinity or negative infinity, a vertical asymptote is present.
Horizontal Asymptotes: These are determined by comparing the degrees of the numerator ($n$) and the denominator ($m$) of the rational function in its simplified form. If $n < m$, the horizontal asymptote is $y = 0$. If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If $n > m$, there is no horizontal asymptote.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the rational function may have an oblique asymptote. This is found by performing polynomial long division.
Limits: Limits are used to analyze the behavior of a function as it approaches a certain value, which is essential in determining the existence of asymptotes.
Rational Functions: A rational function is a ratio of two polynomials. It can have vertical, horizontal, or oblique asymptotes depending on the relationship between the degrees of the numerator and denominator.
