Find the Asymptotes f(x)=(x+6)/(x^2-64)

Solution:

Step 1:

Determine the values of $x$ for which $\frac{x+6}{x^2-64}$ does not exist. These are $x=-8$ and $x=8$.

Step 2:

As $x$ approaches $-8$ from the left, $\frac{x+6}{x^2-64}$ tends towards negative infinity, and from the right, it tends towards positive infinity. This implies a vertical asymptote at $x=-8$.

Step 3:

Similarly, as $x$ approaches $8$ from the left, $\frac{x+6}{x^2-64}$ tends towards negative infinity, and from the right, it tends towards positive infinity, indicating a vertical asymptote at $x=8$.

Step 4:

Compile a list of vertical asymptotes: $x=-8$ and $x=8$.

Step 5:

Examine the general form of 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. The horizontal asymptote rules 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 possibly an oblique asymptote.

Step 6:

Identify the degrees $n$ and $m$. Here, $n=1$ and $m=2$.

Step 7:

Given that $n<m$, the horizontal asymptote is the x-axis, which is $y=0$.

Step 8:

There is no oblique asymptote as the degree of the numerator ($n$) is not greater than the degree of the denominator ($m$).

Step 9:

List all asymptotes for the function:

Vertical Asymptotes: $x=-8$, $x=8$ Horizontal Asymptote: $y=0$ No Oblique Asymptotes

Knowledge Notes:

To find the asymptotes of a rational function like $f(x)=\frac{x+6}{x^2-64}$, one must understand the following concepts:

1. Vertical Asymptotes: These occur where the function is undefined, which is typically where the denominator is zero. 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 ($m$). 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: If the degree of the numerator is one more than the degree of the denominator, the function may have an oblique asymptote. This is found by performing polynomial long division.

4. Behavior Around Asymptotes: To determine the behavior of the function around its vertical asymptotes, one must look at the limits of the function as $x$ approaches the asymptote values from the left and right.

5. Limits: The concept of a limit is used to analyze the behavior of a function as it approaches a certain point. Limits are essential in determining both vertical and horizontal asymptotes.

In the given problem, the function has two vertical asymptotes at $x=-8$ and $x=8$, and one horizontal asymptote at $y=0$. There are no oblique asymptotes because the degree of the numerator is less than the degree of the denominator.