Precalculus Examples

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

Step 1

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

$x = 1$

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

Simplify the expression.

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Step 6.1.1

Factor using the perfect square rule.

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Step 6.1.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 6.1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 6.1.1.3

Rewrite the polynomial.

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

Step 6.1.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 6.1.2

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

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Step 6.1.2.1

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

$\frac{(x - 1) (x - 1)}{x - 1}$

Step 6.1.2.2

Cancel the common factors.

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Step 6.1.2.2.1

Multiply by $1$ .

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

Step 6.1.2.2.2

Cancel the common factor.

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

Step 6.1.2.2.3

Rewrite the expression.

$\frac{x - 1}{1}$

Step 6.1.2.2.4

Divide $x - 1$ by $1$ .

$x - 1$

Step 6.2

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

$y = x - 1$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 1$

Step 8