Precalculus Examples

$r (x) = \frac{x^{3} - 2 x^{2} - 3 x}{x - 3}$

Step 1

Find where the expression $\frac{x^{3} - 2 x^{2} - 3 x}{x - 3}$ is undefined.

$x = 3$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Consider the 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.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

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

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

Step 4

Find $n$ and $m$ .

$n = 3$

$m = 1$

Step 5

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

Factor $x$ out of $x^{3} - 2 x^{2} - 3 x$ .

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Step 6.1.1.1.1

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2} - 2 x^{2} - 3 x}{x - 3}$

Step 6.1.1.1.2

Factor $x$ out of $- 2 x^{2}$ .

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

Step 6.1.1.1.3

Factor $x$ out of $- 3 x$ .

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

Step 6.1.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 2 x)$ .

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

Step 6.1.1.1.5

Factor $x$ out of $x (x^{2} - 2 x) + x \cdot - 3$ .

$\frac{x (x^{2} - 2 x - 3)}{x - 3}$

Step 6.1.1.2

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 6.1.1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 6.1.1.2.2

Write the factored form using these integers.

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

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

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Cancel the common factor of $x - 3$ .

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Step 6.1.2.1.1

Cancel the common factor.

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

Step 6.1.2.1.2

Divide $x (x + 1)$ by $1$ .

$x (x + 1)$

Step 6.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 1$

Step 6.1.2.3

Simplify the expression.

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Step 6.1.2.3.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1$

Step 6.1.2.3.2

Multiply $x$ by $1$ .

$x^{2} + x$

Step 6.2

The oblique asymptote is the polynomial portion of the long division result.

$y = x^{2} + x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x^{2} + x$

Step 8