Find the Asymptotes (t^2-2t)/(t^4-16)
The given problem involves identifying the asymptotes of the function (t^2-2t)/(t^4-16). Asymptotes are lines to which a curve approaches but never touches or intersects. There are usually two types of asymptotes for rational functions: vertical and horizontal (or oblique/slant for certain cases). In this context, the question is asking to determine if the given rational function has any such lines, and if it does, to identify their equations.
To find vertical asymptotes, one would typically look for values of t that make the denominator equal to zero, provided that these values do not also zero out the numerator at the same time, as this would imply a hole in the graph rather than an asymptote.
To find horizontal asymptotes, one would examine the limits of the function as t approaches positive and negative infinity. If the degrees of the polynomials in the numerator and denominator are the same, the horizontal asymptote can often be found by comparing the coefficients of the highest degree terms. If the degree of the numerator is greater, the function may not have a horizontal asymptote; it could have an oblique asymptote instead.
The question does not explicitly ask for oblique asymptotes, but part of the process often involves determining whether the function could have such an asymptote based on the degrees of the numerator and the denominator. If necessary, this would involve long division of the polynomials to find an equation in the form of y = mx + b, where m and b are constants.
$\frac{t^{2} - 2 t}{t^{4} - 16}$
Answer
Solution:
Step 1:
Determine the values of $t$ that make the function $\frac{t^{2} - 2t}{t^{4} - 16}$ undefined. These are $t = -2$ and $t = 2$.
Step 2:
As $t$ approaches $-2$ from the left, $\frac{t^{2} - 2t}{t^{4} - 16}$ tends towards positive infinity, and as $t$ approaches $-2$ from the right, it tends towards negative infinity. Therefore, $t = -2$ is a vertical asymptote.
Step 3:
To find horizontal asymptotes for a function $R(t) = \frac{at^{n}}{bt^{m}}$, 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 are no horizontal asymptotes, but there might be an oblique asymptote.
Step 4:
Calculate the degrees $n$ and $m$. Here, $n = 2$ and $m = 4$.
Step 5:
Since $n < m$, the horizontal asymptote of the function is the x-axis, which is $y = 0$.
Step 6:
There are no oblique asymptotes since the degree of the numerator ($n$) is less than the degree of the denominator ($m$).
Step 7:
The complete set of asymptotes for the function is as follows:
- Vertical Asymptotes: $t = -2$
- 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 values. Here are some key points to consider:
Vertical Asymptotes: These occur at values of $x$ (or $t$ in this case) where the denominator of the function is zero, and the numerator is not zero at the same points. The function will approach infinity or negative infinity near these points.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and denominator (denoted as $n$ and $m$ respectively). If the degree of the numerator is less than the degree of the denominator ($n < m$), the horizontal asymptote is the x-axis ($y = 0$). If the degrees are equal ($n = m$), the horizontal asymptote is the line $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator. If the numerator's degree is greater ($n > m$), there are no horizontal asymptotes.
Oblique Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$). In such cases, one can perform polynomial long division to find the equation of the oblique asymptote.
Undefined Points: The function is undefined where the denominator equals zero. These points are critical for determining vertical asymptotes.
Behavior at Infinity: To find horizontal asymptotes, one must consider the behavior of the function as $x$ (or $t$) goes to positive or negative infinity. If the function approaches a constant value, that value is the horizontal asymptote.
In the given problem, the rational function has a degree of 2 in the numerator and 4 in the denominator, indicating that there will be a horizontal asymptote at $y = 0$ and no oblique asymptotes. The vertical asymptotes are at the points where the denominator is zero, which are $t = -2$ and $t = 2$. However, since the numerator also becomes zero at $t = 2$, it is not a vertical asymptote, leaving $t = -2$ as the only vertical asymptote.
