Find the Asymptotes f(x)=(x+6)/(x^2-64)
The question is asking for the determination of the asymptotes of the given function f(x) = (x+6)/(x^2-64). An asymptote is a line that the graph of a function approaches but never actually reaches as the independent variable (in this case, x) approaches either infinity or a certain point where the function is undefined.
To find the vertical asymptotes, one typically looks for values of x that make the denominator of the function equal to zero, since these are points where the function goes to infinity. Horizontal asymptotes, in contrast, are found by evaluating the behavior of the function as x goes to positive or negative infinity.
The function presents a rational expression where the numerator is a linear polynomial and the denominator is a quadratic polynomial, suggesting that you might expect both vertical asymptote(s) due to factors in the denominator possibly equating to zero, and potentially a horizontal asymptote depending on the leading terms of the numerator and denominator as x approaches infinity.
$f \left(\right. x \left.\right) = \frac{x + 6}{x^{2} - 64}$
Answer
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:
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 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:
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 $x$.
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.
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.
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.
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.
