Find the Asymptotes (2x)/(x^3-7x^2)
The question asks to identify the asymptotes of the given function f(x) = (2x)/(x^3 - 7x^2). An asymptote of a function is a line that the graph of the function approaches but does not actually reach as x approaches infinity or minus infinity or at points where the function is undefined. There are typically three types of asymptotes to consider: horizontal, vertical, and oblique (slant). For rational functions like the one provided, vertical asymptotes are found where the denominator is zero (indicative of a division by zero), and horizontal or oblique asymptotes are determined by the behavior of the function as x approaches infinity or minus infinity. The question requires analyzing the denominator for zeros and the limits of the function at infinity to determine the different types of asymptotes.
$\frac{2 x}{x^{3} - 7 x^{2}}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which $\frac{2x}{x^3 - 7x^2}$ does not exist. These are $x = 0$ and $x = 7$.
Step 2:
Examine the limit of $\frac{2x}{x^3 - 7x^2}$ as $x$ approaches $0$ from the left and right. Since the function approaches $\infty$ from the left and $-\infty$ from the right, $x = 0$ is a vertical asymptote.
Step 3:
Examine the limit of $\frac{2x}{x^3 - 7x^2}$ as $x$ approaches $7$ from the left and right. Since the function approaches $-\infty$ from the left and $\infty$ from the right, $x = 7$ is a vertical asymptote.
Step 4:
Compile a list of vertical asymptotes, which are $x = 0$ and $x = 7$.
Step 5:
Review the criteria for horizontal asymptotes in a 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:
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:
Identify $n$ and $m$ for the given function. Here, $n = 1$ and $m = 3$.
Step 7:
Since $n < m$, the horizontal asymptote is the x-axis, which is $y = 0$.
Step 8:
Conclude that there are no oblique asymptotes because the degree of the numerator is less than the degree of the denominator.
Step 9:
Summarize all the asymptotes of the function:
- Vertical Asymptotes: $x = 0$ and $x = 7$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
Asymptotes are lines that a graph of a function approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique (also known as slant).
Vertical Asymptotes: These occur at values of $x$ where the function is undefined and the limits of the function approach $\infty$ or $-\infty$. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These occur when the values of $y$ approach a constant as $x$ approaches $\infty$ or $-\infty$. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and the denominator ($m$) in a 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. In such cases, you can find the oblique asymptote by performing polynomial long division.
Limits: To determine the behavior of a function near its asymptotes, limits are used. The limit of a function as $x$ approaches a particular value gives us information about the function's behavior around that value.
Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials. The properties of its asymptotes can be analyzed by looking at the degrees of the numerator and denominator polynomials.
