Precalculus Examples

$f (x) = \frac{x^{3}}{{(x - 1)}^{2}}$

Step 1

Find where the expression $\frac{x^{3}}{{(x - 1)}^{2}}$ is undefined.

$x = 1$

Step 2

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 3

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 4

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

Factor using the perfect square rule.

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Step 5.1.1

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

$\frac{x^{3}}{x^{2} - 2 x + 1^{2}}$

Step 5.1.2

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 5.1.3

Rewrite the polynomial.

$\frac{x^{3}}{x^{2} - 2 \cdot x \cdot 1 + 1^{2}}$

Step 5.1.4

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

$\frac{x^{3}}{{(x - 1)}^{2}}$

Step 5.2

Expand ${(x - 1)}^{2}$ .

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Step 5.2.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$\frac{x^{3}}{(x - 1) (x - 1)}$

Step 5.2.2

Apply the distributive property.

$\frac{x^{3}}{x (x - 1) - 1 (x - 1)}$

Step 5.2.3

Apply the distributive property.

$\frac{x^{3}}{x \cdot x + x \cdot - 1 - 1 (x - 1)}$

Step 5.2.4

Apply the distributive property.

$\frac{x^{3}}{x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1}$

Step 5.2.5

Reorder $x$ and $- 1$ .

$\frac{x^{3}}{x \cdot x - 1 x - 1 x - 1 \cdot - 1}$

Step 5.2.6

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

$\frac{x^{3}}{x \cdot x - 1 x - 1 x - 1 \cdot - 1}$

Step 5.2.7

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

$\frac{x^{3}}{x \cdot x - 1 x - 1 x - 1 \cdot - 1}$

Step 5.2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{3}}{x^{1 + 1} - 1 x - 1 x - 1 \cdot - 1}$

Step 5.2.9

Add $1$ and $1$ .

$\frac{x^{3}}{x^{2} - 1 x - 1 x - 1 \cdot - 1}$

Step 5.2.10

Multiply $- 1$ by $- 1$ .

$\frac{x^{3}}{x^{2} - 1 x - 1 x + 1}$

Step 5.2.11

Subtract $1 x$ from $- 1 x$ .

$\frac{x^{3}}{x^{2} - 2 x + 1}$

Step 5.3

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 5.4

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x^{2}$ .

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 5.5

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 2 x^{2} + x$

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

Step 5.7

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

Step 5.8

Pull the next terms from the original dividend down into the current dividend.

$x$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

Step 5.9

Divide the highest order term in the dividend $2 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

Step 5.10

Multiply the new quotient term by the divisor.

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

$2 x^{2}$

$4 x$

$2$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 4 x + 2$

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

$2 x^{2}$

$4 x$

$2$

Step 5.12

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$2$

$x^{2}$

$2 x$

$1$

$x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{3}$

$2 x^{2}$

$x$

$2 x^{2}$

$x$

$0$

$2 x^{2}$

$4 x$

$2$

$3 x$

$2$

Step 5.13

The final answer is the quotient plus the remainder over the divisor.

$x + 2 + \frac{3 x - 2}{x^{2} - 2 x + 1}$

Step 5.14

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

$y = x + 2$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 2$

Step 7