the curve (x^2-2x+2)/(x-1) has an oblique asymptote. what is the equation of this asymptote
To find the equation of the oblique asymptote of the curve \( y = \frac{x^2 - 2x + 2}{x - 1} \), we can perform polynomial long division or use synthetic division to divide the numerator by the denominator.
Performing polynomial long division:
\[
\begin{array}{r|lll}
& x-1 & x-1 & \\
\hline
x^2-2x+2 \,|\, & x^2 & -2x & +2 \\
& -x^2 & +x^2 & \\
\hline
& & -x & \\
& & -x & +x \\
\hline
& & & 2 \\
\end{array}
\]
Therefore, the oblique asymptote of the curve is given by the equation \( y = x - 1 + \frac{2}{x - 1} \).