Precalculus Examples

y = \frac{x}{x^{2 - 4}}

$y = \frac{x}{x^{2 - 4}}$

Step 1

Find where the expression

\frac{x}{x^{2 - 4}}

$\frac{x}{x^{2 - 4}}$ is undefined.

x = 0

$x = 0$

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 expression.

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Step 4.1.1

Multiply the numerator by the reciprocal of the denominator.

x \cdot x^{2}

$x \cdot x^{2}$

Step 4.1.2

Multiply

x

$x$ by

x^{2}

$x^{2}$ by adding the exponents.

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Step 4.1.2.1

Multiply

x

$x$ by

x^{2}

$x^{2}$ .

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Step 4.1.2.1.1

Raise

x

$x$ to the power of

1

$1$ .

x^{1} x^{2}

$x^{1} x^{2}$

Step 4.1.2.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

x^{1 + 2}

$x^{1 + 2}$

x^{1 + 2}

$x^{1 + 2}$

Step 4.1.2.2

Add

1

$1$ and

2

$2$ .

x^{3}

$x^{3}$

x^{3}

$x^{3}$

x^{3}

$x^{3}$

Step 4.2

Since there is no polynomial portion 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