Precalculus Examples

$y = \frac{2 x^{2} - 9 x - 5}{x - 5}$

Step 1

Find where the expression $\frac{2 x^{2} - 9 x - 5}{x - 5}$ is undefined.

$x = 5$

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 by grouping.

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Step 6.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = - 9$ .

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Step 6.1.1.1.1

Factor $- 9$ out of $- 9 x$ .

$\frac{2 x^{2} - 9 (x) - 5}{x - 5}$

Step 6.1.1.1.2

Rewrite $- 9$ as $1$ plus $- 10$

$\frac{2 x^{2} + (1 - 10) x - 5}{x - 5}$

Step 6.1.1.1.3

Apply the distributive property.

$\frac{2 x^{2} + 1 x - 10 x - 5}{x - 5}$

Step 6.1.1.1.4

Multiply $x$ by $1$ .

$\frac{2 x^{2} + x - 10 x - 5}{x - 5}$

Step 6.1.1.2

Factor out the greatest common factor from each group.

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Step 6.1.1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + x) - 10 x - 5}{x - 5}$

Step 6.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x + 1) - 5 (2 x + 1)}{x - 5}$

Step 6.1.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$\frac{(2 x + 1) (x - 5)}{x - 5}$

Step 6.1.2

Cancel the common factor of $x - 5$ .

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Step 6.1.2.1

Cancel the common factor.

$\frac{(2 x + 1) (x - 5)}{x - 5}$

Step 6.1.2.2

Divide $2 x + 1$ by $1$ .

$2 x + 1$

Step 6.2

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

$y = 2 x + 1$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = 2 x + 1$

Step 8