Find the Asymptotes (4x)/(x^2-7x)
The question is asking for the identification of the asymptotes of the function f(x) = (4x)/(x^2-7x). Asymptotes are lines to which a function approaches infinitely closely but never touches or intersects. There are two types of asymptotes to consider:
Vertical asymptotes, which occur where the denominator of the function is equal to zero and the function is undefined.
Horizontal or oblique (slant) asymptotes, which describe the behavior of the function as x approaches positive or negative infinity.
The task involves determining where the function is undefined due to division by zero and analyzing the limit of the function as x tends toward infinity or negative infinity to identify any horizontal or slant asymptote.
$\frac{4 x}{x^{2} - 7 x}$
Answer
Solution:
Step 1:
Determine the values for which the function $\frac{4x}{x^2 - 7x}$ does not exist. These are $x = 0$ and $x = 7$.
Step 2:
Observe the behavior of the function as $x$ approaches $7$. As $x$ approaches $7$ from the left, $\frac{4x}{x^2 - 7x}$ approaches $-\infty$, and from the right, it approaches $\infty$. Therefore, $x = 7$ 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 degree of the numerator and $m$ is the degree of 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, $n = 1$ and $m = 2$.
Step 5:
Since the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, given by $y = 0$.
Step 6:
Conclude that there is no oblique asymptote since 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 Asymptotes: $x = 7$
- Horizontal Asymptotes: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
Asymptotes are lines that a graph approaches but never actually touches or crosses. There are three types of asymptotes: vertical, horizontal, and oblique.
Vertical Asymptotes: These occur at values of $x$ where the function is undefined and typically result from setting the denominator of a rational function to zero. The function will approach infinity or negative infinity as it gets close to these points from one side or the other.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator of the rational function.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. The oblique asymptote can be found by performing polynomial long division on the rational function.
In the given problem, we are dealing with a rational function where the degree of the numerator is less than the degree of the denominator, which means there will be a horizontal asymptote at $y = 0$. Since the degree of the numerator is not greater than the degree of the denominator, there is no oblique asymptote.
