Does $\frac{\sin x}{x}$ have a horizontal asymptote?

Question

I recently came across a definition of an asymptote of a function as follows:

1. The function gets infinitely close to the asymptote.
2. The function does not cross the asymptote.

This made me wonder whether $\frac{\sin x}{x}$ has an asymptote at y = 0, as it fits condition 1 but does not fit 2. I looked it up, and different sources give different results.

Does $\frac{\sin x}{x}$ have a horizontal asymptote at y=0? And if so, what is the problem with this definition? Is it an oversimplification or completely incorrect? And if not, why is that so, as it fits the definition: "a line that a function approaches as x approaches infinity"?

(Also, it would be great if you could list a few reputable sources, but you don't have to.)

Answer 1

The definition I'm familiar to, regarding real valued function $f:\Bbb R \to \Bbb R$ is the following

We say that $y = l$, where $l\in \Bbb R$, is an horizontal asymptote for $f(x)$ if $$\lim_{x\to +\infty} f(x) = l$$

Analogous formulation for $-\infty$. As you can see this represent the first one of your condition, but the second isn't necessary. Here's graph of the function $f(x) = \frac{\sin x}{x}$

And as you verify by yourself the function intersect $y = 0$ (multiple times) but horizontal asymptote still exist!

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P.S. This example is quite importante because one could think that this behaviour on infinite closeness should happen at least monotonically, instead as the name suggest we are interested in this asymptotic approach.