Find the Asymptotes (7x-1)/x
The problem is asking for the identification of any asymptotes in the graph of the rational function (7x - 1)/x. Asymptotes are lines that the graph of a function approaches but does not actually reach, acting as a kind of boundary for the behavior of the graph at infinity or near certain points. Specifically, the request is to determine if there are any vertical or horizontal asymptotes (or possibly oblique asymptotes, which are diagonal lines) for the given function. Vertical asymptotes typically occur where the denominator of a rational function is zero (indicating a division by zero, which is undefined), while horizontal or oblique asymptotes relate to the end behavior of the function as x approaches infinity or negative infinity. The process involves analyzing the function's algebraic form and applying calculus concepts, if necessary, to find these lines.
$\frac{7 x - 1}{x}$
Answer
Solution:
Step 1:
Identify the points where the function $\frac{7x-1}{x}$ is not defined. This occurs when $x = 0$.
Step 2:
Analyze the rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ represents the degree of the polynomial in the numerator and $m$ represents the degree of the polynomial in the denominator. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
If $n = m$, the horizontal asymptote is given by $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 3:
Determine the values of $n$ and $m$. For our function, $n = 1$ and $m = 1$.
Step 4:
Given that $n = m$, we find the horizontal asymptote using the formula $y = \frac{a}{b}$. Here, $a = 7$ and $b = 1$, so the horizontal asymptote is $y = 7$.
Step 5:
Since the degree of the numerator is not greater than the degree of the denominator, there are no oblique asymptotes for this function.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptotes: $x = 0$
- Horizontal Asymptotes: $y = 7$
- No Oblique Asymptotes
Knowledge Notes:
Asymptotes are lines that a graph approaches but never actually touches or crosses. They can be vertical, horizontal, or oblique (slant).
Vertical asymptotes occur at values of $x$ where the function is undefined, typically where the denominator of a rational function is zero.
Horizontal asymptotes are found by comparing the degrees of the numerator and denominator of a rational function:
If the degree of the numerator ($n$) is less than the degree of the denominator ($m$), 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, but there may be an oblique asymptote.
Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They can be found using polynomial long division.
In the given problem, the function $\frac{7x-1}{x}$ has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 7$. There are no oblique asymptotes because the degrees of the numerator and denominator are equal.
