Find the Asymptotes f(x)=(4x^3+18x^2+18x)/(x^3+3x^2)
The question is asking to determine the asymptotes of the given function, which is a rational function (a ratio of two polynomials). Asymptotes are lines that the graph of the function approaches but never actually reaches. There are two types of asymptotes for rational functions: vertical asymptotes, which occur at values of x where the denominator equals zero (as long as those points aren’t also zeros of the numerator, which would indicate a hole in the graph rather than an asymptote), and horizontal or oblique (slant) asymptotes, which describe the end behavior of the function as x goes to positive or negative infinity. To find the vertical asymptotes, you would typically set the denominator equal to zero and solve for x. To find horizontal asymptotes, if they exist, you would compare the degrees of the numerator and the denominator. For an oblique asymptote, if the degree of the numerator is exactly one more than the degree of the denominator, you might perform long division or another method to determine the equation of the asymptote. The question necessitates the application of calculus and algebra to effectively determine these asymptotes.
$f \left(\right. x \left.\right) = \frac{4 x^{3} + 18 x^{2} + 18 x}{x^{3} + 3 x^{2}}$
Answer
Solution:
Step 1:
Determine the values for which the function $\frac{4x^3 + 18x^2 + 18x}{x^3 + 3x^2}$ does not exist. These are $x = -3$ and $x = 0$.
Step 2:
Analyze the behavior of the function as $x$ approaches $0$. The function tends towards $-\infty$ when approaching $0$ from the left and towards $\infty$ when approaching from the right. Hence, $x = 0$ is a vertical asymptote.
Step 3:
Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the highest power of $x$ in the numerator and $m$ is the highest power in 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, but there may be an oblique asymptote.
Step 4:
Identify the degrees $n$ and $m$ for the given function. Here, $n = 3$ and $m = 3$.
Step 5:
Since $n$ equals $m$, the horizontal asymptote is given by $y = \frac{a}{b}$, where $a = 4$ and $b = 1$. Therefore, the horizontal asymptote is $y = 4$.
Step 6:
An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of asymptotes for the function:
- Vertical Asymptote: $x = 0$
- Horizontal Asymptote: $y = 4$
- No Oblique Asymptotes
Step 8:
Knowledge Notes:
To find the asymptotes of a rational function, one must follow these steps:
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero (and not the numerator at the same time). To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator ($n$) and the denominator ($m$) of the 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: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, perform polynomial long division to find the linear equation that the graph of the function approaches as $x$ goes to infinity.
Behavior at Asymptotes: To determine how the graph behaves near the vertical asymptotes, one should analyze the limits of the function as it approaches the asymptote from the left and right.
Nonexistence of Asymptotes: There are cases where no horizontal or oblique asymptotes exist, particularly when the degree of the numerator is greater than the degree of the denominator by more than one.
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