Find the Asymptotes y=(-3x^6+5x+3)/(9x^6+6x+4)

$y = \frac{- 3 x^{6} + 5 x + 3}{9 x^{6} + 6 x + 4}$

Step 1

Find where the expression $\frac{- 3 x^{6} + 5 x + 3}{9 x^{6} + 6 x + 4}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

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 $m$ .

$n = 6$

$m = 6$

Step 5

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = - 3$ and $b = 9$ .

$y = - \frac{1}{3}$

Step 6

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = - \frac{1}{3}$

No Oblique Asymptotes

Step 8

Find the Asymptotes y=(-3x^6+5x+3)/(9x^6+6x+4)

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