Find the Asymptotes f(x)=(x^2-4)/(16x-x^4)
The task requires the identification of the asymptotes of the given function, which is a rational function of the form f(x)=(x²-4)/(16x-x⁴). An asymptote refers to a line that the graph of a function approaches but does not touch as the values of x approach infinity or negative infinity, or at certain specific points in the domain of the function. There can be vertical asymptotes, which occur at values of x where the denominator of the fraction approaches zero but the numerator does not, and horizontal or oblique (slant) asymptotes, which describe the behavior of the graph as x goes to positive or negative infinity. The challenge here is to find these lines by analyzing the mathematical properties of the function and its limits.
$f \left(\right. x \left.\right) = \frac{x^{2} - 4}{16 x - x^{4}}$
Answer
Solution:
Step 1:
Identify the values for which $\frac{x^2 - 4}{16x - x^4}$ is not defined. These are $x = 0$ and $x = 2^{2/3}$.
Step 2:
Analyze the behavior of $\frac{x^2 - 4}{16x - x^4}$ 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 $\frac{x^2 - 4}{16x - x^4}$ as $x$ approaches $2^{2/3}$. It diverges to positive infinity from the left and negative infinity from the right, confirming another vertical asymptote at $x = 2^{2/3}$.
Step 4:
Compile a list of vertical asymptotes found: $x = 0$ and $x = 2^{2/3}$.
Step 5:
Consider a general 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 rules 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, and one should check for an oblique asymptote.
Step 6:
Determine the degrees $n$ and $m$ for the given function. Here, $n = 2$ and $m = 4$.
Step 7:
Since $n < m$, the horizontal asymptote is the x-axis, given by $y = 0$.
Step 8:
An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.
Step 9:
Summarize all asymptotes:
- Vertical Asymptotes: $x = 0$, $x = 2^{2/3}$
- Horizontal Asymptote: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
The process of finding asymptotes for a rational function involves several steps:
Vertical Asymptotes: These occur where the function is undefined, typically where the denominator is zero. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are determined by comparing the degrees of the numerator and denominator ($n$ and $m$, respectively). 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. 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. To find an oblique asymptote, divide the numerator by the denominator using polynomial long division.
Behavior at Asymptotes: To confirm a vertical asymptote, check the limits of the function as $x$ approaches the value from the left and right. If the function tends to infinity or negative infinity, it confirms the existence of a vertical asymptote.
Limits and Continuity: Understanding limits is crucial for determining the behavior of functions near their asymptotes. Continuity plays a role in understanding where a function is defined and its behavior around points of discontinuity.
In the given problem, we have a rational function where the degree of the numerator is less than the degree of the denominator, which means there will be a horizontal asymptote at $y = 0$ and no oblique asymptote. Vertical asymptotes are found by setting the denominator to zero and solving for $x$.
