Recall that division by zero is not defined. Therefore, the rational function is undefined where (x-1)^2=0.
(x-1)^2=0 ⇔ x=1
The function is not defined when x=1. This means we have either a hole or a vertical asymptote. Let's now simplify the function.
After simplifying, the factor x-1 is still in the denominator. This indicates that the line x=1 is a vertical asymptote.
Extra
Horizontal Asymptotes
We are also able to find horizontal asymptotes in rational functions.
y=p(x)/q(x)
The degree of p(x) is greater than the degree of q(x)
There are no horizontal asymptotes
The degree of q(x) is greater than the degree of p(x)
y=0 is a horizontal asymptote
The degrees of p(x) and q(x) are equal
y= a b is a horizontal asymptote, where a and b are the leading coefficients of p(x) and q(x), respectively
Let's now consider the given function.
y=x^1-1/x^2-2x+1
Since the degree of the denominator is greater than the degree of the numerator, the line y=0 is a horizontal asymptote.
