Find the Asymptotes f(x)=(x^2+x-6)/(x^2-4x-21)

Solution:

Step 1:

Determine the values of $x$ that make the function $\frac{x^2+x-6}{x^2-4x-21}$ undefined. These are $x=-3$ and $x=7$.

Step 2:

Observe the behavior of $\frac{x^2+x-6}{x^2-4x-21}$ as $x$ approaches $7$. It tends towards $-\mathrm{\infty }$ from the left and $\mathrm{\infty }$ from the right, indicating $x=7$ is a vertical asymptote.

Step 3:

For a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator:

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

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

• If $n>m$, there are no horizontal asymptotes, but there may be an oblique asymptote.

Step 4:

Identify the degrees $n$ and $m$ of the numerator and denominator, respectively. Here, $n=2$ and $m=2$.

Step 5:

Since the degrees $n$ and $m$ are equal, the horizontal asymptote is given by $y=\frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. In this case, $y=1$.

Step 6:

There are no oblique asymptotes since the degree of the numerator is not greater than the degree of the denominator.

Step 7:

Compile the list of all asymptotes for the function:

• Vertical Asymptotes: $x=7$
• Horizontal Asymptotes: $y=1$
• No Oblique Asymptotes

Knowledge Notes:

To find the asymptotes of a rational function $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, follow these steps:

1. Vertical Asymptotes: These occur at the values of $x$ that make the denominator $Q(x)$ equal to zero, provided that the numerator $P(x)$ does not also equal zero at these points. To find them, solve $Q(x)=0$.

2. Horizontal Asymptotes: These depend on the degrees of the numerator and denominator polynomials.

   • If the degree of $P(x)$ is less than the degree of $Q(x)$, the horizontal asymptote is $y=0$.

   • If the degrees are equal, the horizontal asymptote is $y=\frac{a}{b}$, where $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$, respectively.

   • If the degree of $P(x)$ is greater than the degree of $Q(x)$, there is no horizontal asymptote.

3. Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the function may have an oblique asymptote. To find it, perform polynomial long division of $P(x)$ by $Q(x)$.

4. Behavior Around Vertical Asymptotes: To determine how the function behaves around its vertical asymptotes, analyze the limits of $f(x)$ as $x$ approaches the vertical asymptote from the left and right.

5. Combining Asymptotes: The set of all asymptotes provides a sketch of the function's behavior at infinity and near critical points where the function is undefined.