Find the Asymptotes 4/(2x^2-11x+5)
The question is asking for the determination of the asymptotes of the function f(x) = 4/(2x^2-11x+5). An asymptote is a line that the graph of a function approaches but never actually reaches. The problem requires finding both vertical and horizontal (or possibly oblique) asymptotes, if they exist. For vertical asymptotes, one would typically look for values of x that make the denominator zero, and for horizontal or oblique asymptotes, one would analyze the behavior of the function as x approaches infinity or negative infinity. The question necessitates the application of calculus and algebraic techniques to identify lines that the graph of the given rational function will approach at great distances from the origin on the Cartesian plane.
$\frac{4}{2 x^{2} - 11 x + 5}$
Answer
Solution:
Step:1
Determine the values of $x$ that make the denominator of $\frac{4}{2x^2 - 11x + 5}$ equal to zero. These values are $x = \frac{1}{2}$ and $x = 5$.
Step:2
As $x$ approaches $\frac{1}{2}$ from the left, $\frac{4}{2x^2 - 11x + 5}$ tends towards positive infinity, and as $x$ approaches $\frac{1}{2}$ from the right, it tends towards negative infinity. Therefore, $x = \frac{1}{2}$ is a vertical asymptote.
Step:3
As $x$ approaches $5$ from the left, $\frac{4}{2x^2 - 11x + 5}$ tends towards negative infinity, and as $x$ approaches $5$ from the right, it tends towards positive infinity. Thus, $x = 5$ is a vertical asymptote.
Step:4
Compile a list of the vertical asymptotes: $x = \frac{1}{2}$ and $x = 5$.
Step:5
Examine 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. The rules for horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is the x-axis, $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step:6
Identify the values of $n$ and $m$. Here, $n = 0$ and $m = 2$.
Step:7
Given that $n < m$, the horizontal asymptote is the x-axis, which is the line $y = 0$.
Step:8
There is no oblique asymptote since the degree of the numerator ($n$) is less than the degree of the denominator ($m$).
Step:9
Summarize all asymptotes found:
Vertical Asymptotes: $x = \frac{1}{2}$ and $x = 5$ Horizontal Asymptotes: $y = 0$ No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the behavior of the function as the variable approaches certain critical points or infinity. Here are the key points to consider:
Vertical Asymptotes: These occur at values of $x$ where the denominator of the rational function is zero, and the function tends towards infinity or negative infinity. To find these, 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 function in its simplified form. 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 degree of the numerator is greater than the degree of the denominator, 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, polynomial long division can be used to find the equation of the oblique asymptote. If the degree of the numerator is less than or equal to the degree of the denominator, as in this case, there are no oblique asymptotes.
Undefined Points: The function is undefined where the denominator is zero, which is also where vertical asymptotes can be found.
Behavior at Infinity: To determine the horizontal asymptote, if any, one must consider the end behavior of the function as $x$ approaches infinity or negative infinity.
Latex Formatting: In the solution, Latex is used to properly format mathematical expressions, ensuring clarity and readability.
