Find the Asymptotes (8+x^4)/(x^2-x^4)

Solution:

Step 1:

Determine the values for which the function $\frac{8+x^4}{x^2-x^4}$ is not defined. These are the values where the denominator equals zero: $x=-1,x=0,x=1$.

Step 2:

Examine the limit of $\frac{8+x^4}{x^2-x^4}$ as $x$ approaches $-1$. It diverges to $-\mathrm{\infty }$ from the left and to $\mathrm{\infty }$ from the right, indicating a vertical asymptote at $x=-1$.

Step 3:

Observe the behavior of the function as $x$ approaches $0$. The function tends to $\mathrm{\infty }$ from both sides, suggesting a vertical asymptote at $x=0$.

Step 4:

Investigate the limit as $x$ approaches $1$. The function trends towards $\mathrm{\infty }$ from the left and $-\mathrm{\infty }$ from the right, confirming a vertical asymptote at $x=1$.

Step 5:

Compile a list of all vertical asymptotes: $x=-1,0,1$.

Step 6:

Consider a general rational function $R(x)=\frac{a{x}^n}{b{x}^m}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator. The rules for horizontal asymptotes are as follows:

1. If $n<m$, the horizontal asymptote is $y=0$.

2. If $n=m$, the horizontal asymptote is $y=\frac{a}{b}$.

3. If $n>m$, there is no horizontal asymptote (an oblique asymptote may exist).

Step 7:

Identify the degrees $n$ and $m$ of the numerator and denominator, respectively. In this case, $n=4$ and $m=4$.

Step 8:

Since $n=m$, the horizontal asymptote is determined by the leading coefficients of the numerator and denominator. With $a=1$ and $b=-1$, the horizontal asymptote is $y=-1$.

Step 9:

An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.

Step 10:

Summarize the set of all asymptotes:

Vertical Asymptotes: $x=-1,0,1$ Horizontal Asymptote: $y=-1$ No Oblique Asymptotes

Knowledge Notes:

To find the asymptotes of a rational function, we follow these steps:

1. Vertical Asymptotes: These occur at the values of $x$ that make the denominator zero (provided that the numerator is not zero at the same points). To find them, set the denominator equal to zero and solve for $x$.

2. Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the 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.

3. 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, perform polynomial long division or synthetic division to find the slant asymptote equation.

4. 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 approaches $\mathrm{\infty }$ or $-\mathrm{\infty }$ from either side, it confirms the presence of a vertical asymptote.

5. Limits and Continuity: Understanding limits is crucial for finding asymptotes. If the limit of a function as $x$ approaches a certain value is infinite, it suggests the presence of a vertical asymptote.

6. Polynomial Division: When checking for oblique asymptotes, polynomial division is used to divide the numerator by the denominator when the degree of the numerator is greater than the degree of the denominator.

7. Leading Coefficients: The leading coefficients of the numerator and denominator play a significant role in determining the equation of the horizontal asymptote when the degrees of the numerator and denominator are equal.