Question

determine the equation of the slant asymptote of the rational function: $y = \frac{x^{2}-4x + 11}{x - 3}$
$y=x + 1$
$y=x + 7$
$y=x - 7$
$y=x - 1$

Explanation:

Step1: Perform polynomial long - division

Divide $x^{2}-4x + 11$ by $x - 3$.
We know that $x^{2}-4x + 11=x(x - 3)-x+11=x(x - 3)-(x - 3)+8$. So, $\frac{x^{2}-4x + 11}{x - 3}=x - 1+\frac{8}{x - 3}$.

Step2: Determine the slant asymptote

As $x\to\pm\infty$, the term $\frac{8}{x - 3}\to0$. The equation of the slant asymptote is given by the non - remainder part of the long - division result.

Answer:

$y=x - 1$