Find the Asymptotes y=(5x^3-2x+1)/(2x^3+8x^2-x+3)

$y = \frac{5 x^{3} - 2 x + 1}{2 x^{3} + 8 x^{2} - x + 3}$

Step 1

Find where the expression $\frac{5 x^{3} - 2 x + 1}{2 x^{3} + 8 x^{2} - x + 3}$ is undefined.

$x = - 4.20381904$

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 $m$ .

$n = 3$

$m = 3$

Step 4

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = 5$ and $b = 2$ .

$y = \frac{5}{2}$

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: $x = - 4.20381904$

Horizontal Asymptotes: $y = \frac{5}{2}$

No Oblique Asymptotes

Step 7