# How to find the asymptotes of #f(x) = (x^2+x-2)/( x^3-3x^2+2x)#?

##### 1 Answer

#### Explanation:

Start by factoring both numerator and denominator and cancelling common factors:

#f(x) = (x^2+x-2)/(x^3-3x^2+2x) = ((x+2)color(red)(cancel(color(black)((x-1)))))/(x(x-2)color(red)(cancel(color(black)((x-1))))) = ((x+2))/(x(x-2))#

with exclusion

Notice that when

#(color(blue)(1)+2)/(color(blue)(1)(color(blue)(1)-2)) = 3/(-1) = -3#

So

The remaining values of

Finally, since the degree of the numerator is greater than the denominator,

It has no slant (oblique) asymptotes. Such slant asymptotes can only occur if the degree of the numerator is

Here's a graph of

graph{(y-(x+2)/(x(x-2)))(0.9999x-2)(0.9999x+0.0001y) = 0 [-8.71, 11.29, -5.76, 4.24]}