How to find asymptotes of $y=ax+b+\frac{c+\sin x}{x}$

Question

How can we find the asymptotes of $y=ax+b+\frac{c+\sin x}{x}$?

What kind of asymptotes do you want to work on first? What have you tried? — rschwieb, Dec 31 '12 at 16:31

score 1 · Accepted Answer · answered Dec 31 '12 at 16:31

1

HINT: Notice that when $|x|$ is very large, the fraction $$\frac{c+\sin x}x$$ is very small. (Why?) Thus, when $|x|$ is very large,

$$y=ax+b+\frac{c+\sin x}x\approx ax+b\;.$$

answered Dec 31 '12 at 16:31

Brian M. Scott

score 1 · Answer 2 · answered Dec 31 '12 at 16:31

1

There is an oblique asymptote with equation $y=ax+b$.

If $c\neq 0$, there is also a vertical asymptote with equation $x=0$.

answered Dec 31 '12 at 16:31

Julien

score 0 · Answer 3 · answered Dec 31 '12 at 16:31

0

Hint: For vertical asymptotes look at $x=0$. What happens there?

For all other asymptotes first compute $\lim_{x\to \pm \infty}\frac{f(x)}{x}$ where $f(x)=ax+b+(c+\sin x)/x$

answered Dec 31 '12 at 16:31

Nameless

3 Answers3