Find the Asymptotes (t^2-2t)/(t^4-16)

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 $R(t)=\frac{a{t}^n}{b{t}^m}$, 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:

1. 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.

2. 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.

3. 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.

4. Undefined Points: The function is undefined where the denominator equals zero. These points are critical for determining vertical asymptotes.

5. 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.