Find the Asymptotes (X^2+2X-3)/(X-1)

Solution:

Step 1:

Identify the values for which the function $\frac{x^2+2x-3}{x-1}$ is not defined. This occurs when $x=1$.

Step 2:

Determine the locations of vertical asymptotes, which are typically where the function is undefined due to division by zero. In this case, there are no vertical asymptotes.

Step 3:

Examine the degrees of the numerator and denominator in the rational function $R(x)=\frac{a{x}^n}{b{x}^m}$. The rules for horizontal asymptotes 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 are no horizontal asymptotes, but possibly an oblique asymptote.

Step 4:

Calculate the degrees of the numerator and denominator. Here, $n=2$ and $m=1$.

Step 5:

Since the degree of the numerator is greater than the degree of the denominator ($n>m$), there are no horizontal asymptotes.

Step 6:

To find the oblique asymptote, perform polynomial long division.

Step 6.1:

First, simplify the given expression.

Step 6.1.1:

Factor the numerator $x^2+2x-3$ using the AC method.

Step 6.1.1.1:

Look for two integers whose product equals $c$ and whose sum equals $b$. In this case, find integers with a product of $-3$ and a sum of $2$. These integers are $-1$ and $3$.

Step 6.1.1.2:

Express the factored form with these integers as $\frac{(x-1)(x+3)}{x-1}$.

Step 6.1.2:

Eliminate the common factor from the numerator and denominator.

Step 6.1.2.1:

Remove the common factor to get $\frac{\cancel{(x-1)}(x+3)}{\cancel{x-1}}$.

Step 6.1.2.2:

Divide the remaining polynomial $x+3$ by $1$ to get $x+3$.

Step 6.2:

The quotient from the division gives the equation of the oblique asymptote, which is $y=x+3$.

Step 7:

Combine all the findings to list the asymptotes of the function. There are no vertical or horizontal asymptotes, but there is an oblique asymptote at $y=x+3$.

Knowledge Notes:

• Asymptotes are lines that a graph approaches but does not actually reach. They can be vertical, horizontal, or oblique (slant).

• Vertical asymptotes occur where the function is undefined due to division by zero.

• Horizontal asymptotes are determined by the degrees of the numerator ($n$) and denominator ($m$) in a rational function. If $n<m$, the horizontal asymptote is $y=0$; if $n=m$, it is $y=\frac{a}{b}$; if $n>m$, there are no horizontal asymptotes.

• Oblique asymptotes occur when the degree of the numerator is one more than the degree of the denominator. They can be found by performing polynomial long division.

• Polynomial long division is a method used to divide polynomials, similar to the long division of numbers. The result can provide the equation of an oblique asymptote.

• Factoring using the AC method involves finding two numbers that multiply to give the product of the coefficient of $x^2$ (A) and the constant term (C), and add up to the coefficient of $x$ (B). This is used to factor quadratic expressions.

• Cancellation is the process of simplifying a fraction by eliminating common factors from the numerator and the denominator.