Find the Asymptotes f(x)=(x^2+1)/(x^2-4)
The given problem asks for the identification of the asymptotes of the function f(x) = (x^2+1)/(x^2-4). Asymptotes are lines that the graph of a function approaches as the independent variable (in this case, x) approaches infinity or some constant value. Specifically, this problem is requesting the determination of where the function f(x) behaves in such a manner that it becomes arbitrarily close to a straight line either as x becomes very large (horizontal asymptote), very small (horizontal asymptote), or near certain values where the function is undefined (vertical asymptotes). It usually involves analyzing the behavior of the function at extreme values of x and where the denominator equals zero.
$f \left(\right. x \left.\right) = \frac{x^{2} + 1}{x^{2} - 4}$
Answer
Solution:
Step 1:
Identify the values for which the function $\frac{x^2 + 1}{x^2 - 4}$ is not defined. These are $x = -2$ and $x = 2$.
Step 2:
Analyze the behavior of $\frac{x^2 + 1}{x^2 - 4}$ as $x$ approaches $-2$. As $x$ approaches $-2$ from the left, the function approaches $\infty$, and from the right, it approaches $-\infty$. Therefore, $x = -2$ is a vertical asymptote.
Step 3:
Examine the behavior of $\frac{x^2 + 1}{x^2 - 4}$ as $x$ approaches $2$. As $x$ approaches $2$ from the left, the function approaches $-\infty$, and from the right, it approaches $\infty$. Thus, $x = 2$ is a vertical asymptote.
Step 4:
Compile a list of the vertical asymptotes, which are $x = -2$ and $x = 2$.
Step 5:
Consider a general 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 6:
Determine the values of $n$ and $m$. For our function, $n = 2$ and $m = 2$.
Step 7:
Since $n = m$, the horizontal asymptote is given by $y = \frac{a}{b}$, where $a = 1$ and $b = 1$. Hence, the horizontal asymptote is $y = 1$.
Step 8:
There is no oblique asymptote for this function because the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize all asymptotes of the function:
- Vertical Asymptotes: $x = -2$, $x = 2$
- Horizontal Asymptote: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function like $f(x) = \frac{x^2 + 1}{x^2 - 4}$, we follow these steps:
Vertical Asymptotes: These occur at the values of $x$ that make the denominator zero, as long as those values do not also make the numerator zero (which would indicate a hole instead of an asymptote). 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 ($m$) of the rational function:
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: These occur when the degree of the numerator is exactly one more than the degree of the denominator. To find an oblique asymptote, perform polynomial long division or synthetic division to find the slant asymptote equation.
Behavior near Asymptotes: To determine how the function behaves near its vertical asymptotes, we can analyze the limits as $x$ approaches the asymptote values from the left and right.
Listing Asymptotes: After determining the vertical and horizontal (or oblique) asymptotes, we list them to provide a complete description of the function's end behavior.
In the given problem, we have a rational function where the numerator and denominator are both of degree 2, and the leading coefficients are both 1. Therefore, we have a horizontal asymptote at $y = 1$ and vertical asymptotes at $x = -2$ and $x = 2$. There are no oblique asymptotes because the degree of the numerator is not greater than the degree of the denominator.
