Question

How do you find the asymptotes for `y=(3x-2) / (x+1)`?

Answer 1

There is a horizontal asymptote: `y = 3`

There is a vertical asymptote: `x = -1`

Explanation:

You can rewrite the expression.

`frac{3x-2}{x + 1} = frac{3x + 3 - 5}{x + 1}`

`= frac{3x + 3}{x + 1} + frac{-5}{x + 1}`

`= 3 - frac{5}{x + 1}`

From this, you can see that

`lim_{x -> oo} frac{3x-2}{x + 1} = lim_{x -> oo} (3 - frac{5}{x + 1}) = 3`

Similarly,

`lim_{x -> -oo} frac{3x-2}{x + 1} = lim_{x -> -oo} (3 - frac{5}{x + 1}) = 3`

There is a horizontal asymptote: `y = 3`

You can also see that `x = -1` results in division by zero. When `x` approaches `-1` from the left, the denominator will become infinisimally less than zero, while the numerator is negative. So,

`lim_{x -> -1^-} frac{3x-2}{x + 1} = oo`

Similarly,

`lim_{x -> -1^+} frac{3x-2}{x + 1} = -oo`

There is a vertical asymptote: `x = -1`

Below is a graph for your reference.
graph{(3x-2)/(x + 1) [-20, 20, -10, 10]}