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

##### 1 Answer

There is a horizontal asymptote:

There is a vertical asymptote:

#### 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:

You can also see that

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

Similarly,

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

There is a vertical asymptote:

Below is a graph for your reference.

graph{(3x-2)/(x + 1) [-20, 20, -10, 10]}