Question

An oblique asymptote to the curve $y=x+e^{-x}\sin x$ is

• A. $y=x+\frac{\pi }{2}$
• B. $y=x+\frac{2}{\pi }$
• C. $y=x+e$
• D. $y=x$

Answer 1

The correct option is D $y=x$

If $y=mx+c$ is an oblique asymptote to the curve $y=f\left(x\right)$, then $x\to +\infty$ or $x\to -\infty$ we have
$m=\underset{x\to -\infty }{lim}\left(\frac{y}{x}\right)=\underset{x\to -\infty }{lim}\frac{x+e^{-x}\sin x}{x}=\underset{x\to -\infty }{lim}(1+e^{-x}\frac{\sin x}{x})=1$
and $c={lim}_{x\to \infty }(y-mx)={lim}_{x\to \infty }(x+e^{-x}\sin x-x)={lim}_{x\to \infty }{e}^{-x}\sin x=0$
$\therefore$ Oblique asymptote is $y=mx+c$ i.e. $y=x$ $(\because c=0)$