Examples

y = \frac{x + 1}{x^{2} - 1}

$y = \frac{x + 1}{x^{2} - 1}$

Step 1

Find where the expression

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

$\frac{x + 1}{x^{2} - 1}$ is undefined.

x = - 1, x = 1

$x = - 1, x = 1$

Step 2

Since

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

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

\to

$\to$

- \infty

$- \infty$ as

x

$x$

\to

$\to$

1

$1$ from the left and

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

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

\to

$\to$

\infty

$\infty$ as

x

$x$

\to

$\to$

1

$1$ from the right, then

x = 1

$x = 1$ is a vertical asymptote.

x = 1

$x = 1$

Step 3

Consider the rational function

R (x) = \frac{a x^{n}}{b x^{m}}

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

n

$n$ is the degree of the numerator and

m

$m$ is the degree of the denominator.

1. If

n < m

$n < m$ , then the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

2. If

n = m

$n = m$ , then the horizontal asymptote is the line

y = \frac{a}{b}

$y = \frac{a}{b}$ .

3. If

n > m

$n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

Find

n

$n$ and

m

$m$ .

n = 1

$n = 1$

m = 2

$m = 2$

Step 5

Since

n < m

$n < m$ , the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

y = 0

$y = 0$

Step 6

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

Vertical Asymptotes:

x = 1

$x = 1$

Horizontal Asymptotes:

y = 0

$y = 0$

No Oblique Asymptotes

Step 8

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