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As far as I have learned a function has a horizontal asymptote $y = k$ if and only if $$\lim_{x \to \infty} f(x) = k$$ or $$\lim_{x \to -\infty} f(x) = k$$

Now, for a constant function $$f(x) = c$$ we have $$\lim_{x \to \infty} f(x) = \lim_{x \to -\infty} f(x) = c$$

Does this mean that all constant functions have horizontal asymptotes, or is this definition not the one commonly used?

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It really depends on how you define asymptotes, and there was much discussion about this matter in this question. It really comes down to whether or not we should have

$$f(x)=c$$

At only a countably infinite amount of times as $x\to\pm\infty$. But for all intended purposes, I think stating a constant function is asymptotic to itself, or indeed, that any function is asymptotic to itself, is, though not very useful, acceptably correct by standard definitions.

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The function is defined everywhere. It would be an asymptote if the limit exists AND the function is not defined for that y value.

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  • $\begingroup$ That's not true. We can say $\sqrt{x^2+1}$ is asymptotic to $|x|$ as $x\to\pm\infty$, for example. More over, the function does not have to be defined everywhere. $$\lim_{x\to\infty}\frac1{\sqrt x}=0$$ $\endgroup$ – Simply Beautiful Art Nov 12 '16 at 14:48

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