Pre-Algebra Examples

\frac{(x - 3) (x + 2)}{(x - 1) (x - 3)}

$\frac{(x - 3) (x + 2)}{(x - 1) (x - 3)}$

Step 1

Find where the expression

\frac{x + 2}{x - 1}

$\frac{x + 2}{x - 1}$ is undefined.

x = 1

$x = 1$

Step 2

Consider the rational function

R (x) = \frac{a x^{n}}{b x^{m}}

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

n

$n$ is the degree of the numerator and

m

$m$ is the degree of the denominator.

1. If

n < m

$n < m$ , then the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

2. If

n = m

$n = m$ , then the horizontal asymptote is the line

y = \frac{a}{b}

$y = \frac{a}{b}$ .

3. If

n > m

$n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 3

Find

n

$n$ and

m

$m$ .

n = 1

$n = 1$

m = 1

$m = 1$

Step 4

Since

n = m

$n = m$ , the horizontal asymptote is the line

y = \frac{a}{b}

$y = \frac{a}{b}$ where

a = 1

$a = 1$ and

b = 1

$b = 1$ .

y = 1

$y = 1$

Step 5

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 6

This is the set of all asymptotes.

Vertical Asymptotes:

x = 1

$x = 1$

Horizontal Asymptotes:

y = 1

$y = 1$

No Oblique Asymptotes

Step 7

\frac{(x - 3) (x + 2)}{(x - 1) (x - 3)}

$\frac{(x - 3) (x + 2)}{(x - 1) (x - 3)}$

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