Question

An Oblique asymptote to the curve $y=\frac{x^2+2x-1}{x}$ is

• A. $y=-x+2$
• B. $y=x+2$
• C. $y=-x-2$
• D. $y=x-2$

Answer 1

Answer: (B)

$y=x+2$

If $y=mx+c$ is an oblique asymptote to the curve y=f(x), then $x\to +\infty$ or $x\to -\infty$
we have $m=\underset{x\to \pm \infty }{lim}\left(\frac{y}{x}\right)=\underset{x\to \pm \infty }{lim}\frac{x^2+2x-1}{x^2}=1$ and
$c={lim}_{x\to \infty }(y-mx)={lim}_{x\to \infty }(f\left(x\right)-mx)$
$=\underset{x\to \infty }{lim}(\frac{x^2+2x-1}{x}-x)$ $(\because m=1)$
$=\underset{x\to \infty }{lim}\left(\frac{2x-1}{x}\right)=2$
$\therefore$ Oblique asymptote is $y=mx+c=x+2$