Precalculus Examples

$f (x) = \frac{x^{3} + 1}{x^{2} - 4}$

Step 1

Find where the expression $\frac{x^{3} + 1}{x^{2} - 4}$ is undefined.

$x = - 2, x = 2$

Step 2

Since $\frac{x^{3} + 1}{x^{2} - 4}$ $\to$ $- \infty$ as $x$ $\to$ $- 2$ from the left and $\frac{x^{3} + 1}{x^{2} - 4}$ $\to$ $\infty$ as $x$ $\to$ $- 2$ from the right, then $x = - 2$ is a vertical asymptote.

$x = - 2$

Step 3

Since $\frac{x^{3} + 1}{x^{2} - 4}$ $\to$ $- \infty$ as $x$ $\to$ $2$ from the left and $\frac{x^{3} + 1}{x^{2} - 4}$ $\to$ $\infty$ as $x$ $\to$ $2$ from the right, then $x = 2$ is a vertical asymptote.

$x = 2$

Step 4

List all of the vertical asymptotes:

$x = - 2, 2$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

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

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the expression.

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Step 8.1.1

Simplify the numerator.

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Step 8.1.1.1

Rewrite $1$ as $1^{3}$ .

$\frac{x^{3} + 1^{3}}{x^{2} - 4}$

Step 8.1.1.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$\frac{(x + 1) (x^{2} - x \cdot 1 + 1^{2})}{x^{2} - 4}$

Step 8.1.1.3

Simplify.

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Step 8.1.1.3.1

Multiply $- 1$ by $1$ .

$\frac{(x + 1) (x^{2} - x + 1^{2})}{x^{2} - 4}$

Step 8.1.1.3.2

One to any power is one.

$\frac{(x + 1) (x^{2} - x + 1)}{x^{2} - 4}$

Step 8.1.2

Simplify the denominator.

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Step 8.1.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 8.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 8.2

Expand $(x + 1) (x^{2} - x + 1)$ .

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Step 8.2.1

Apply the distributive property.

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

Step 8.2.2

Apply the distributive property.

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

Step 8.2.3

Apply the distributive property.

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

Step 8.2.4

Apply the distributive property.

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

Step 8.2.5

Apply the distributive property.

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

Step 8.2.6

Remove parentheses.

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

Step 8.2.7

Reorder $x$ and $- 1$ .

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

Step 8.2.8

Reorder $x$ and $1$ .

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

Step 8.2.9

Remove parentheses.

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

Step 8.2.10

Multiply $- 1$ by $x$ .

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

Step 8.2.11

Raise $x$ to the power of $1$ .

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

Step 8.2.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.2.13

Add $1$ and $2$ .

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

Step 8.2.14

Factor out negative.

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

Step 8.2.15

Raise $x$ to the power of $1$ .

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

Step 8.2.16

Raise $x$ to the power of $1$ .

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

Step 8.2.17

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.2.18

Add $1$ and $1$ .

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

Step 8.2.19

Multiply $x$ by $1$ .

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

Step 8.2.20

Multiply $x^{2}$ by $1$ .

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

Step 8.2.21

Multiply $- 1$ by $1$ .

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

Step 8.2.22

Multiply $1$ by $1$ .

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

Step 8.2.23

Move $x$ .

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

Step 8.2.24

Move $- x^{2}$ .

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

Step 8.2.25

Subtract $x^{2}$ from $x^{2}$ .

$\frac{x^{3} + 0 + x - x + 1}{(x + 2) (x - 2)}$

Step 8.2.26

Add $x^{3}$ and $0$ .

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

Step 8.2.27

Subtract $x$ from $x$ .

$\frac{x^{3} + 0 + 1}{(x + 2) (x - 2)}$

Step 8.2.28

Add $x^{3}$ and $0$ .

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

Step 8.3

Expand $(x + 2) (x - 2)$ .

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Step 8.3.1

Apply the distributive property.

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

Step 8.3.2

Apply the distributive property.

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

Step 8.3.3

Apply the distributive property.

$\frac{x^{3} + 1}{x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2}$

Step 8.3.4

Reorder $x$ and $- 2$ .

$\frac{x^{3} + 1}{x \cdot x - 2 x + 2 x + 2 \cdot - 2}$

Step 8.3.5

Raise $x$ to the power of $1$ .

$\frac{x^{3} + 1}{x \cdot x - 2 x + 2 x + 2 \cdot - 2}$

Step 8.3.6

Raise $x$ to the power of $1$ .

$\frac{x^{3} + 1}{x \cdot x - 2 x + 2 x + 2 \cdot - 2}$

Step 8.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3} + 1}{x^{1 + 1} - 2 x + 2 x + 2 \cdot - 2}$

Step 8.3.8

Add $1$ and $1$ .

$\frac{x^{3} + 1}{x^{2} - 2 x + 2 x + 2 \cdot - 2}$

Step 8.3.9

Multiply $2$ by $- 2$ .

$\frac{x^{3} + 1}{x^{2} - 2 x + 2 x - 4}$

Step 8.3.10

Add $- 2 x$ and $2 x$ .

$\frac{x^{3} + 1}{x^{2} + 0 - 4}$

Step 8.3.11

Subtract $4$ from $0$ .

$\frac{x^{3} + 1}{x^{2} - 4}$

Step 8.4

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 8.5

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 8.6

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{3}$

$0$

$4 x$

Step 8.7

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - 4 x$

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{3}$

$0$

$4 x$

Step 8.8

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{3}$

$0$

$4 x$

Step 8.9

Pull the next term from the original dividend down into the current dividend.

$x$

$x^{2}$

$0 x$

$4$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{3}$

$0$

$4 x$

$1$

Step 8.10

The final answer is the quotient plus the remainder over the divisor.

$x + \frac{4 x + 1}{x^{2} - 4}$

Step 8.11

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

$y = x$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 2, 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x$

Step 10