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

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What kind of asymptotes do you want to work on first? What have you tried? – rschwieb Dec 31 '12 at 16:31
up vote 1 down vote accepted

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,

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

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

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

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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 $f(x)=ax+b+(c+\sin x)/x$

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