Precalculus Examples

$f (x) = \frac{9 x^{6}}{7 x}$

Step 1

Find where the expression $\frac{9 x^{6}}{7 x}$ is undefined.

$x = 0$

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 $m$ .

$n = 6$

$m = 1$

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

Cancel the common factor of $x^{6}$ and $x$ .

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Step 6.1.1

Factor $x$ out of $9 x^{6}$ .

$\frac{x (9 x^{5})}{7 x}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $x$ out of $7 x$ .

$\frac{x (9 x^{5})}{x \cdot 7}$

Step 6.1.2.2

Cancel the common factor.

$\frac{x (9 x^{5})}{x \cdot 7}$

Step 6.1.2.3

Rewrite the expression.

$\frac{9 x^{5}}{7}$

Step 6.2

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

$7$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 6.3

Divide the highest order term in the dividend $9 x^{5}$ by the highest order term in divisor $7$ .

$\frac{9 x^{5}}{7}$

$7$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 6.4

Multiply the new quotient term by the divisor.

$\frac{9 x^{5}}{7}$

$7$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$9 x^{5}$

Step 6.5

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{5}$

$\frac{9 x^{5}}{7}$

$7$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$9 x^{5}$

Step 6.6

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

$\frac{9 x^{5}}{7}$

$7$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$9 x^{5}$

$0$

Step 6.7

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

$\frac{9 x^{5}}{7}$

$7$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$9 x^{5}$

$0$

$0 x^{4}$

Step 6.8

Since the remander is $0$ , the final answer is the quotient.

$\frac{9 x^{5}}{7}$

Step 6.9

Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8