Find the Asymptotes f(x)=(2-x-x^3)/(x^3-1)

Solution:

Step:1 Determine the values of $x$ for which the function $\frac{2-x-x^3}{x^3-1}$ is not defined. This occurs when the denominator equals zero. Thus, $x=1$.

Step:2 Identify the vertical asymptotes, which are typically located where the function is not defined and the limit approaches infinity. In this case, there are no vertical asymptotes.

Step:3 For a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, with $n$ as the highest power in the numerator and $m$ as the highest power in the denominator, follow these rules:

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 are no horizontal asymptotes; instead, look for an oblique asymptote.

Step:4 Calculate the degrees $n$ and $m$ of the numerator and denominator, respectively. For our function, $n=3$ and $m=3$.

Step:5 Given that $n=m$, the horizontal asymptote can be found using $y=\frac{a}{b}$. The leading coefficients are $a=-1$ and $b=1$, hence $y=-1$.

Step:6 An oblique asymptote is not present since the degree of the numerator is not greater than the degree of the denominator. Therefore, we do not have any oblique asymptotes.

Step:7 Compile the complete list of asymptotes for the function:

• No Vertical Asymptotes
• Horizontal Asymptote: $y=-1$
• No Oblique Asymptotes

Knowledge Notes:

• Rational Functions: A rational function is a function that can be expressed as the quotient of two polynomials. It is of the form $R(x)=\frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\ne 0$.

• Asymptotes: Asymptotes are lines that the graph of a function approaches as $x$ or $y$ goes to infinity. There are three types of asymptotes: vertical, horizontal, and oblique (or slant).

• Vertical Asymptotes: These occur at values of $x$ where the function goes to infinity. They are found by setting the denominator equal to zero and solving for $x$.

• Horizontal Asymptotes: These are horizontal lines that the graph approaches as $x$ goes to infinity. For a rational function $R(x)=\frac{a{x}^n}{b{x}^m}$, if $n<m$, the horizontal asymptote is $y=0$; if $n=m$, it's $y=\frac{a}{b}$; and if $n>m$, there are no horizontal asymptotes.

• Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator, the graph may have an oblique asymptote. This is found by performing polynomial long division or synthetic division.

• Undefined Points: A function is undefined at points where the denominator is zero. These points are not part of the function's domain.

• Leading Coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest power.

Understanding these concepts is crucial for analyzing the behavior of rational functions and determining their asymptotes.