Find the Asymptotes y=(-3x^5+5x+4)/(6x^5+2x+5)

$y = \frac{- 3 x^{5} + 5 x + 4}{6 x^{5} + 2 x + 5}$

Step 1

Find where the expression $\frac{- 3 x^{5} + 5 x + 4}{6 x^{5} + 2 x + 5}$ is undefined.

$x = - 0.88365467$

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 $m$ .

$n = 5$

$m = 5$

Step 4

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = - 3$ and $b = 6$ .

$y = - \frac{1}{2}$

Step 5

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 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 0.88365467$

Horizontal Asymptotes: $y = - \frac{1}{2}$

No Oblique Asymptotes

Step 7