Find the Asymptotes 1/(x^2-2x)
The task is to determine the asymptotes of the function f(x) = 1/(x^2 - 2x). Asymptotes are lines that the graph of a function approaches but does not actually reach as the magnitude of x increases to infinity or decreases to negative infinity. There are typically two kinds of asymptotes to consider for rational functions like this one; vertical and horizontal (and sometimes oblique).
Vertical asymptotes occur at values of x which make the denominator equal to zero and the function undefined, provided the numerator does not also equal zero at those points.
Horizontal asymptotes are found by analyzing the behavior of the function as x approaches infinity and negative infinity. This involves comparing the degrees of the polynomials in the numerator and the denominator or using limits.
The question requires examining the function to determine where these asymptotes are by following the appropriate mathematical procedures.
$\frac{1}{x^{2} - 2 x}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which the function $f(x) = \frac{1}{x^2 - 2x}$ is not defined. These occur where the denominator equals zero. Solving $x^2 - 2x = 0$, we find $x = 0$ and $x = 2$.
Step 2:
Examine the behavior of $f(x)$ as $x$ approaches $0$. As $x$ approaches $0$ from the left, $f(x)$ approaches positive infinity, and from the right, it approaches negative infinity. Hence, $x = 0$ is a vertical asymptote.
Step 3:
Analyze the behavior of $f(x)$ as $x$ approaches $2$. As $x$ approaches $2$ from the left, $f(x)$ approaches negative infinity, and from the right, it approaches positive infinity. Consequently, $x = 2$ is a vertical asymptote.
Step 4:
Compile a list of the vertical asymptotes we've found: $x = 0$ and $x = 2$.
Step 5:
To find horizontal asymptotes for a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ and $m$ are the degrees of the numerator and denominator respectively:
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
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:
Identify the degrees $n$ and $m$ for our function. Here, $n = 0$ and $m = 2$.
Step 7:
Since the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, which is $y = 0$.
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 for the function:
- Vertical Asymptotes: $x = 0$ and $x = 2$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a function, we need to understand the following concepts:
Vertical Asymptotes occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes are found by comparing the degrees of the numerator ($n$) and the denominator ($m$) in a rational function $R(x) = \frac{ax^n}{bx^m}$. Depending on the relationship between $n$ and $m$, the horizontal asymptote can be $y = 0$, $y = \frac{a}{b}$, or nonexistent.
Oblique Asymptotes (also known as slant asymptotes) may occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, long division of the polynomial can be used to find the equation of the oblique asymptote.
Behavior of the Function near the vertical asymptotes is important to determine the direction in which the function approaches infinity.
Limits are a fundamental tool in finding asymptotes, as they describe the behavior of a function as it approaches a certain point from the left or right.
Rational Functions are ratios of two polynomials. The asymptotic behavior of rational functions is largely determined by the degrees and coefficients of these polynomials.
