Find the Asymptotes f(x)=(5x)/(x^3-7x^2)
The question asks to determine the asymptotes of the function f(x) = (5x) / (x^3 - 7x^2). Asymptotes are lines that the graph of a function approaches but never actually reaches as x moves towards infinity or minus infinity, or as the function approaches a vertical or horizontal line. There are typically three types of asymptotes to consider: vertical, horizontal, and oblique (or slant) asymptotes.
Vertical asymptotes occur at points where the function is undefined, typically where the denominator of a rational function is zero. Horizontal asymptotes reflect the behavior of a function as x approaches infinity or negative infinity and are found by analyzing the end behavior of the function. Oblique or slant asymptotes may occur when the degree of the polynomial in the numerator is exactly one higher than the degree of the polynomial in the denominator, and they are found by long division or synthetic division of the polynomials.
The question essentially requires knowledge of rational functions and their limits. It is asking you to analyze the function to find any points where the function cannot exist (vertical asymptotes), where it stabilizes at a constant value (horizontal asymptotes), or where it approaches a straight line (oblique asymptotes). It involves calculus concepts, specifically limits, and may also require algebraic manipulation, such as polynomial long division, to resolve cases where horizontal or oblique asymptotes might occur.
$f \left(\right. x \left.\right) = \frac{5 x}{x^{3} - 7 x^{2}}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which $\frac{5x}{x^3 - 7x^2}$ does not exist. These are $x = 0$ and $x = 7$.
Step 2:
Analyze the behavior of $\frac{5x}{x^3 - 7x^2}$ as $x$ approaches $0$. It tends towards positive infinity from the left and negative infinity from the right, indicating a vertical asymptote at $x = 0$.
Step 3:
Examine the limit of $\frac{5x}{x^3 - 7x^2}$ as $x$ approaches $7$. It approaches negative infinity from the left and positive infinity from the right, confirming a vertical asymptote at $x = 7$.
Step 4:
Compile a list of vertical asymptotes: $x = 0$ and $x = 7$.
Step 5:
Review the general form of 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. The horizontal asymptote depends on the relationship between $n$ and $m$:
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; instead, there may be an oblique asymptote.
Step 6:
Identify the degrees $n$ 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:
There is no slant asymptote because the degree of the numerator ($n$) is not greater than the degree of the denominator ($m$).
Step 9:
Summarize the asymptotes of the function:
- Vertical Asymptotes: $x = 0$ and $x = 7$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a function, especially a rational function, one must understand the behavior of the function as $x$ approaches certain critical values. Here are some key points:
Vertical Asymptotes: These occur at values of $x$ where the denominator of a rational function is zero and the numerator is not zero. The function will approach infinity or negative infinity at these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and denominator ($n$ and $m$ respectively). If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$. If the degrees are equal, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If the numerator's degree is greater, there is no horizontal asymptote.
Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. In such cases, one would perform polynomial long division to find the equation of the oblique asymptote.
Limits: The concept of limits is crucial in determining the behavior of a function as it approaches a certain value, which is fundamental in finding asymptotes.
Rational Functions: A rational function is a ratio of two polynomials. It is written in the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Understanding these concepts allows for the systematic determination of the asymptotes of a given rational function.
