Find the Asymptotes (-x^2+5x-4)/(5x^2+5x-10)

Solution:

Step 1:

Identify the values of $x$ that cause the function $f(x)=\frac{-x^2+5x-4}{5x^2+5x-10}$ to be undefined. These are the roots of the denominator: $x=-2,x=1$.

Step 2:

Examine the behavior of $f(x)$ as $x$ approaches $-2$. As $x\to -2^{-}$, $f(x)\to -\mathrm{\infty }$, and as $x\to -2^{+}$, $f(x)\to \mathrm{\infty }$. Thus, $x=-2$ is a vertical asymptote.

Step 3:

To determine horizontal asymptotes, consider the degrees of the numerator and denominator in the function $R(x)=\frac{a{x}^n}{b{x}^m}$. The 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 (an oblique asymptote may exist).

Step 4:

Calculate the degrees of the numerator and denominator. For the given function, $n=2$ and $m=2$.

Step 5:

Since the degrees are equal ($n=m$), the horizontal asymptote is found using $y=\frac{a}{b}$. Here, $a=-1$ and $b=5$, so the horizontal asymptote is $y=-\frac{1}{5}$.

Step 6:

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

Step 7:

Compile the list of asymptotes:

• Vertical Asymptotes: $x=-2$
• Horizontal Asymptotes: $y=-\frac{1}{5}$
• No Oblique Asymptotes

Knowledge Notes:

To find the asymptotes of a rational function, one must understand the different types of asymptotes:

1. Vertical Asymptotes: These occur at values of $x$ where the function is undefined, typically where the denominator is zero. To find them, set the denominator equal to zero and solve for $x$.

2. Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator ($n$) and denominator ($m$):

   • 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, respectively.

   • 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. The oblique asymptote can be found by performing polynomial long division.

4. Undefined Points: These are specific values of $x$ for which the function does not have a real value. They are found by setting the denominator equal to zero and solving for $x$.

5. Behavior Around Asymptotes: To determine the behavior of the function around vertical asymptotes, one must analyze the limits of the function as $x$ approaches the asymptote from the left and right.

In LaTeX, to write fractions, use the $\frac{numerator}{denominator}$command. For powers, use the caret symbol $\text{Missing superscript or subscript argument}$followed by the exponent in curly braces $$. For example, $x^2$ is written as $x^2$.