Precalculus Examples

$f (x) = \frac{3 (2 x + 5) (2 x - 5)}{3 (2 x + 5)}$

Step 1

Find where the expression $\frac{3 (2 x + 5) (2 x - 5)}{3 (2 x + 5)}$ is undefined.

$x = - \frac{5}{2}$

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

Cancel the common factor of $12 x^{2} - 75$ and $6 x + 15$ .

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Step 6.1.1.1

Factor $3$ out of $12 x^{2}$ .

$\frac{3 (4 x^{2}) - 75}{6 x + 15}$

Step 6.1.1.2

Factor $3$ out of $- 75$ .

$\frac{3 (4 x^{2}) + 3 (- 25)}{6 x + 15}$

Step 6.1.1.3

Factor $3$ out of $3 (4 x^{2}) + 3 (- 25)$ .

$\frac{3 (4 x^{2} - 25)}{6 x + 15}$

Step 6.1.1.4

Cancel the common factors.

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Step 6.1.1.4.1

Factor $3$ out of $6 x$ .

$\frac{3 (4 x^{2} - 25)}{3 (2 x) + 15}$

Step 6.1.1.4.2

Factor $3$ out of $15$ .

$\frac{3 (4 x^{2} - 25)}{3 (2 x) + 3 (5)}$

Step 6.1.1.4.3

Factor $3$ out of $3 (2 x) + 3 (5)$ .

$\frac{3 (4 x^{2} - 25)}{3 (2 x + 5)}$

Step 6.1.1.4.4

Cancel the common factor.

$\frac{3 (4 x^{2} - 25)}{3 (2 x + 5)}$

Step 6.1.1.4.5

Rewrite the expression.

$\frac{4 x^{2} - 25}{2 x + 5}$

Step 6.1.2

Simplify the numerator.

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Step 6.1.2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{{(2 x)}^{2} - 25}{2 x + 5}$

Step 6.1.2.2

Rewrite $25$ as $5^{2}$ .

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

Step 6.1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 5$ .

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

Step 6.1.3

Cancel the common factor of $2 x + 5$ .

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Divide $2 x - 5$ by $1$ .

$2 x - 5$

Step 6.2

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

$y = 2 x - 5$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = 2 x - 5$

Step 8