Find the Asymptotes f(x)=(4x^3+18x^2+18x)/(x^3+3x^2)

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 $-\mathrm{\infty }$ when approaching $0$ from the left and towards $\mathrm{\infty }$ when approaching from the right. Hence, $x=0$ is a vertical asymptote.

Step 3:

Examine the general form of a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, 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:

1. If $n<m$, the horizontal asymptote is $y=0$.

2. If $n=m$, the horizontal asymptote is $y=\frac{a}{b}$.

3. 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:

1. 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 $x$.

2. 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.

3. 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.

4. 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.

5. 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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