Asymptotes of a rational function

Text

Find the asymptotes (vertical and oblique) of the function $y=\dfrac{x^{2}+1}{x-2}$.

Solution

Domain: $x\neq2$. Since $\displaystyle\lim_{x\to2}\dfrac{x^{2}+1}{x-2}=\pm\infty$, the vertical asymptote is $x=2$. Polynomial division: $\dfrac{x^{2}+1}{x-2}=x+2+\dfrac{5}{x-2}$. Since $\dfrac{5}{x-2}\to0$ as $x\to\pm\infty$, the oblique asymptote is $y=x+2$ ($m=1$, $q=2$). $\boxed{\text{vertical asymptote }x=2;\quad \text{oblique }y=x+2}$