Find the Asymptotes (x^2-x-2)/(x^3-2x^2-5x+6)
The question asks for the identification of any asymptotes of the function f(x) = (x^2-x-2)/(x^3-2x^2-5x+6). An asymptote is a line that the graph of the function approaches but never actually reaches as x goes towards infinity or negative infinity, or when x approaches a specific value. There are vertical asymptotes where the denominator of the function approaches zero (and the numerator does not also approach zero at the same points), and there may be horizontal or oblique asymptotes as x approaches infinity, which describe the end behavior of the function. This question involves examining the given rational function to determine where these asymptotes occur.
$\frac{\left(\right. x^{2} - x - 2 \left.\right)}{\left(\right. x^{3} - 2 x^{2} - 5 x + 6 \left.\right)}$
Answer
Solution:
Step 1:
Determine the values of $x$ that make the function $\frac{x^2 - x - 2}{x^3 - 2x^2 - 5x + 6}$ undefined: $x = -2, x = 1, x = 3$.
Step 2:
Analyze the behavior of the function as $x$ approaches $-2$. The function tends to $-\infty$ from the left and $\infty$ from the right, indicating a vertical asymptote at $x = -2$.
Step 3:
Examine the limit as $x$ approaches $1$. The function approaches $-\infty$ from the left and $\infty$ from the right, suggesting a vertical asymptote at $x = 1$.
Step 4:
Investigate the limit as $x$ approaches $3$. The function heads towards $-\infty$ from the left and $\infty$ from the right, confirming a vertical asymptote at $x = 3$.
Step 5:
Compile a list of all vertical asymptotes: $x = -2, 1, 3$.
Step 6:
Consider the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator. The rules for horizontal asymptotes are as follows:
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 (instead, there may be an oblique asymptote).
Step 7:
Identify the degrees $n$ and $m$: $n = 2$, $m = 3$.
Step 8:
Since $n < m$, the horizontal asymptote is the x-axis, given by $y = 0$.
Step 9:
There are no oblique asymptotes, as the degree of the numerator is not greater than the degree of the denominator.
Step 10:
Summarize all asymptotes:
Vertical Asymptotes: $x = -2, 1, 3$ Horizontal Asymptotes: $y = 0$ No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, we need to understand the following concepts:
Vertical Asymptotes: These occur at values of $x$ where the denominator of the rational function is zero (provided that the numerator is not also zero at these points). To find vertical asymptotes, we solve for the roots of the denominator.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator ($n$) and the denominator ($m$) of the rational function. If the degree of the numerator is less than the degree of the denominator ($n < m$), the horizontal asymptote is the x-axis ($y = 0$). If the degrees are equal ($n = m$), the horizontal asymptote is the line $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If the numerator's degree is greater than the denominator's ($n > m$), there is no horizontal asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$). In such cases, we perform polynomial long division to find the equation of the oblique asymptote. If $n$ is not greater than $m$, there are no oblique asymptotes.
Behavior at Asymptotes: To determine the behavior of the function near its vertical asymptotes, we examine the limits of the function as $x$ approaches the asymptote values from the left and right.
Limits: Understanding limits is crucial in analyzing the behavior of a function near its asymptotes. Limits help us determine how the function behaves as it approaches a certain value of $x$.
Rational Functions: A rational function is a fraction of two polynomials. Its asymptotic behavior is determined by the relationship between the degrees of the numerator and denominator polynomials.
