Precalculus Examples

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

Step 1

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

$x = 0$

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 $m$ .

$n = 2$

$m = 1$

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

Cancel the common factor of $x^{2}$ and $x$ .

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Step 6.1.1

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

$\frac{x (2 x)}{x}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

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

Step 6.1.2.2

Factor $x$ out of $x^{1}$ .

$\frac{x (2 x)}{x \cdot 1}$

Step 6.1.2.3

Cancel the common factor.

$\frac{x (2 x)}{x \cdot 1}$

Step 6.1.2.4

Rewrite the expression.

$\frac{2 x}{1}$

Step 6.1.2.5

Divide $2 x$ by $1$ .

$2 x$

Step 6.2

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

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8