Precalculus Examples

f (x) = \frac{x^{3} - 27}{| x - 3 |}

$f (x) = \frac{x^{3} - 27}{| x - 3 |}$

Step 1

Find where the expression

\frac{x^{3} - 27}{| x - 3 |}

$\frac{x^{3} - 27}{| x - 3 |}$ is undefined.

x = 3

$x = 3$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

Since the limit does not exist, there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 4

Find the oblique asymptote using polynomial division.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite

27

$27$ as

3^{3}

$3^{3}$ .

\frac{x^{3} - 3^{3}}{| x - 3 |}

$\frac{x^{3} - 3^{3}}{| x - 3 |}$

Step 4.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula,

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

a = x

$a = x$ and

b = 3

$b = 3$ .

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

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

Step 4.1.3

Simplify.

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Step 4.1.3.1

Move

3

$3$ to the left of

x

$x$ .

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

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

Step 4.1.3.2

Raise

3

$3$ to the power of

2

$2$ .

\frac{(x - 3) (x^{2} + 3 x + 9)}{| x - 3 |}

$\frac{(x - 3) (x^{2} + 3 x + 9)}{| x - 3 |}$

\frac{(x - 3) (x^{2} + 3 x + 9)}{| x - 3 |}

$\frac{(x - 3) (x^{2} + 3 x + 9)}{| x - 3 |}$

\frac{(x - 3) (x^{2} + 3 x + 9)}{| x - 3 |}

$\frac{(x - 3) (x^{2} + 3 x + 9)}{| x - 3 |}$

Step 4.2

Since there is no remainder from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 5

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 6