Find the Asymptotes (5x-8)/(x-6)
The question asks for the identification of the asymptotes of the function f(x) = (5x-8)/(x-6). An asymptote is a line that the graph of a function approaches but does not actually reach. This problem likely involves finding both vertical and horizontal asymptotes, which correspond to values that x can approach but not reach (due to division by zero, for vertical asymptotes), and the limiting behavior of the function as x approaches infinity (for horizontal asymptotes), respectively.
$\frac{5 x - 8}{x - 6}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the function $\frac{5x - 8}{x - 6}$ to be undefined. This occurs when $x = 6$.
Step 2:
Examine 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 $y = 0$.
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 3:
Determine the values of $n$ and $m$. In this case, $n = 1$ and $m = 1$.
Step 4:
Since $n$ equals $m$, the horizontal asymptote can be found using the formula $y = \frac{a}{b}$, where $a = 5$ and $b = 1$. Therefore, the horizontal asymptote is $y = 5$.
Step 5:
An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptote: $x = 6$
- Horizontal Asymptote: $y = 5$
- There are no Oblique Asymptotes.
Step 7:
Knowledge Notes:
To find the asymptotes of a rational function like $\frac{5x - 8}{x - 6}$, one must understand the different types of asymptotes and how they relate to the degrees of the numerator and denominator polynomials.
Vertical Asymptotes: These occur at values of $x$ that make the denominator zero (and the numerator non-zero), as the function approaches infinity or negative infinity at those points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively):
If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
If $n = m$, the horizontal asymptote is found by dividing the leading coefficients of the numerator and the denominator, $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote.
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.
In the given function, since the degree of the numerator and denominator are equal ($n = m$), we find the horizontal asymptote using the leading coefficients ($a$ and $b$). There is no oblique asymptote because the degree of the numerator is not greater than the degree of the denominator. The vertical asymptote is found by setting the denominator equal to zero and solving for $x$.
