Find the Asymptotes (5x^2)/(x^2-3x-4)
The problem presents a rational function (5x^2)/(x^2-3x-4) and asks for its asymptotes. To address this question, one would need to find both the vertical and horizontal (or possibly oblique) asymptotes of the given function. Vertical asymptotes occur where the denominator equals zero and the function is undefined, while horizontal or oblique asymptotes are about the behavior of the function as it approaches infinity or negative infinity. It involves analyzing the degree of the polynomial in the numerator relative to the polynomial in the denominator and applying the appropriate theorems or rules to find the equations of these asymptotes.
$\frac{5 x^{2}}{x^{2} - 3 x - 4}$
Answer
Solution:
Step 1:
Identify the values of $x$ that make the denominator of $\frac{5x^2}{x^2 - 3x - 4}$ equal to zero. These are $x = -1$ and $x = 4$.
Step 2:
Analyze the behavior of $\frac{5x^2}{x^2 - 3x - 4}$ as $x$ approaches $-1$. It tends towards positive infinity from one side and negative infinity from the other, indicating a vertical asymptote at $x = -1$.
Step 3:
Examine the behavior of $\frac{5x^2}{x^2 - 3x - 4}$ as $x$ approaches $4$. It diverges to negative infinity from one side and positive infinity from the other, confirming a vertical asymptote at $x = 4$.
Step 4:
Compile a list of all vertical asymptotes: $x = -1$ and $x = 4$.
Step 5:
Consider 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 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 degrees $n$ and $m$ for the numerator and denominator, respectively. Here, $n = 2$ and $m = 2$.
Step 7:
Since $n$ equals $m$, the horizontal asymptote is given by $y = \frac{a}{b}$, where $a = 5$ and $b = 1$. Thus, the horizontal asymptote is $y = 5$.
Step 8:
Conclude that there are no oblique asymptotes, as the degree of the numerator is not greater than that of the denominator.
Step 9:
Summarize the asymptotes of the function:
Vertical Asymptotes: $x = -1$, $x = 4$ Horizontal Asymptote: $y = 5$ No Oblique Asymptotes
Knowledge Notes:
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero. The function will approach infinity or negative infinity near these points.
Horizontal Asymptotes: These are horizontal lines that the graph of a function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and 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 (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. The oblique asymptote can be found by performing polynomial long division.
Behavior Near Asymptotes: To determine the behavior of the function near a vertical asymptote, one must look at the limits of the function as it approaches the asymptote from the left and the right. If the function approaches infinity on one side and negative infinity on the other, it confirms the presence of a vertical asymptote.
Rational Functions: These are functions expressed as the ratio of two polynomials. The degree of a polynomial is the highest power of $x$ with a non-zero coefficient.
Limits: The concept of a limit is fundamental in calculus and is used to describe the behavior of functions as they approach specific points or infinity.
