Find the Asymptotes f(x)=(7x)/(x^2-6x)

Solution:

Step 1:

Identify the values for which the function $\frac{7x}{x^2-6x}$ does not exist. These are $x=0$ and $x=6$.

Step 2:

Observe the behavior of the function as $x$ approaches $6$. The function tends to negative infinity when approaching $6$ from the left and to positive infinity when approaching from the right. Hence, $x=6$ is a vertical asymptote.

Step 3:

For a general 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 is no horizontal asymptote, but there may be an oblique asymptote.

Step 4:

Determine the values of $n$ and $m$. For our function, $n=1$ and $m=2$.

Step 5:

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

Step 6:

An oblique asymptote is not present since the degree of the numerator ($n$) is not greater than the degree of the denominator ($m$).

Step 7:

Compile the list of asymptotes for the function:

• Vertical Asymptote: $x=6$
• Horizontal Asymptote: $y=0$
• No Oblique Asymptotes

Knowledge Notes:

To find the asymptotes of a rational function, one must understand the different types of asymptotes and how they relate to the function's behavior at various points.

1. Vertical Asymptotes: These occur at values of $x$ where the function is undefined and typically result from setting the denominator equal to zero. The function will approach infinity or negative infinity near these points.

2. Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively). If the degree of the numerator is less than the degree of the denominator ($n<m$), the horizontal asymptote is $y=0$. If the degrees are equal ($n=m$), the horizontal asymptote is $y=\frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If the numerator's degree is greater ($n>m$), 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 ($n=m+1$). In such cases, one must perform long division of the numerator by the denominator to find the equation of the oblique asymptote.

4. Undefined Points: It is important to identify where the function is undefined, as these points often lead to vertical asymptotes.

5. Behavior at Infinity: Understanding the behavior of the function as $x$ approaches infinity or negative infinity can help determine the presence of horizontal or oblique asymptotes.

In the given problem, the function $\frac{7x}{x^2-6x}$ has a vertical asymptote at $x=6$ and a horizontal asymptote at $y=0$. There is no oblique asymptote because the degree of the numerator is less than the degree of the denominator.