Calculus Examples

$f (x) = \frac{x^{3} + 8}{x + 2}$

Step 1

Find where the expression $\frac{x^{3} + 8}{x + 2}$ is undefined.

$x = - 2$

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

Rewrite $8$ as $2^{3}$ .

$\frac{x^{3} + 2^{3}}{x + 2}$

Step 6.1.1.2

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 $b = 2$ .

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

Step 6.1.1.3

Simplify.

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Step 6.1.1.3.1

Multiply $2$ by $- 1$ .

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

Step 6.1.1.3.2

Raise $2$ to the power of $2$ .

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

Step 6.1.2

Cancel the common factor of $x + 2$ .

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Step 6.1.2.1

Cancel the common factor.

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

Step 6.1.2.2

Divide $x^{2} - 2 x + 4$ by $1$ .

$x^{2} - 2 x + 4$

Step 6.2

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

$y = x^{2} - 2 x + 4$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x^{2} - 2 x + 4$

Step 8