Question

The number of asymptotes of the curve $y=\frac{x^2-3x+2}{x^2+3x+2}$ is

Answer 1

Given, $y=\frac{x^2-3x+2}{x^2+3x+2}=\frac{(x-1)(x-2)}{(x+1)(x+2)},$ Domain is $R-\{-1,-2\}$
$\underset{x\to -1^{-}}{lim}y=-\infty ,\underset{x\to -1^{+}}{lim}y=\infty$
$\Rightarrow x=-1$ is a vertical asymptote$\cdots \left(i\right)$
$\underset{x\to -2^{-}}{lim}y=\infty ,\underset{x\to -2^{+}}{lim}y=-\infty$
$\Rightarrow x=-2$ is a vertical asymptote$\cdots \left(ii\right)$
$\underset{x\to \pm \infty }{lim}y=1$
$\Rightarrow y=1$ is a horizontal asymptote$\cdots \left(iii\right)$
Given curve has no inclined asymptotes
$\because \left\{\underset{x\to \pm \infty }{lim}\frac{y}{x}\right\}=\underset{x\to \pm \infty }{lim}\frac{x^2-3x+2}{x(x^2+3x+2)}=0$
Hence the given function has $3$ asymtotes.