Find the Asymptotes f(x)=(5x^2+9)/(x^2-9)
The question is asking for the determination of the asymptotes of the given rational function f(x) = (5x^2 + 9) / (x^2 - 9). An asymptote represents a line that the graph of the function approaches but never actually reaches. Specifically, there are generally two types of asymptotes that might be considered:
Horizontal asymptotes, which occur when x approaches infinity or negative infinity, and the function approaches a constant value.
Vertical asymptotes, which occur at values of x that cause the denominator to be zero (provided that the numerator isn't also zero for these values), which generally means the function is undefined at these points and the value of the function increases or decreases without bound as it approaches these points.
The question requires you to find both types of asymptotes, if they exist, for this particular function.
$f \left(\right. x \left.\right) = \frac{5 x^{2} + 9}{x^{2} - 9}$
Answer
Solution:
Step 1:
Identify the values for which the function $\frac{5x^2 + 9}{x^2 - 9}$ does not exist. These are $x = -3$ and $x = 3$.
Step 2:
Observe that as $x$ approaches $-3$ from the left, $\frac{5x^2 + 9}{x^2 - 9}$ tends towards positive infinity, and from the right, it tends towards negative infinity. Hence, $x = -3$ is a vertical asymptote.
Step 3:
Similarly, as $x$ approaches $3$ from the left, $\frac{5x^2 + 9}{x^2 - 9}$ tends towards negative infinity, and from the right, it tends towards positive infinity. Therefore, $x = 3$ is also a vertical asymptote.
Step 4:
Compile a list of vertical asymptotes: $x = -3$ and $x = 3$.
Step 5:
Analyze the 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 to determine horizontal asymptotes:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes, but there may be an oblique asymptote.
Step 6:
Calculate the degrees $n$ and $m$ for the given function: $n = 2$, $m = 2$.
Step 7:
Since $n$ equals $m$, the horizontal asymptote is found using $y = \frac{a}{b}$, with $a = 5$ and $b = 1$. Therefore, the horizontal asymptote is $y = 5$.
Step 8:
Conclude that there are no oblique asymptotes since the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize the asymptotes of the function:
Vertical Asymptotes: $x = -3$, $x = 3$ Horizontal Asymptote: $y = 5$ No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the behavior of the function as the independent variable approaches certain critical values. There are three types of asymptotes to consider:
Vertical Asymptotes: These occur at values of $x$ where the denominator of the function is zero (provided the numerator is not also zero at those points). The function will approach infinity or negative infinity near these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$ respectively):
If $n < m$, the x-axis ($y = 0$) is the horizontal asymptote.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
If $n > m$, there is no horizontal asymptote.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. The equation of the oblique asymptote can be found using polynomial long division.
In the given problem, we identify vertical asymptotes by finding where the denominator equals zero. We then determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degrees are equal, the horizontal asymptote is the line $y = \frac{a}{b}$. There are no oblique asymptotes in this case because the degree of the numerator is not greater than that of the denominator.
