Is this a horizontal asymptote?

Consider the following function: $$f(x) = \frac{(x-2)^2}{(x^2-1)}$$. To calculate the horizontal asymptote, we take the limit:

$$\displaystyle{\lim_{x \to \infty}} f(x) = 1$$

However, $$f(x)$$ still intersects with $$y = 1$$, namely at the point $$(1.25;1)$$.

So is $$y = 1$$ the horizontal asymptote? Because it still intersects with $$y = 1$$, but the function approaches 1 as x approaches to infinity.

• $y=1$ is horizontal asymptote, you got it right. Consider function $y=\sin \frac{1}{x}$, it has infinitely many intersections with $y=0$ which is horizontal asymptote – Vasya Mar 28 at 16:08
• Ah, that's really interesting. Thanks for the example! I assume vertical asymptotes can't be intersected, but horizontal asymptotes can right? – Stallmp Mar 28 at 16:14
• Yes, that's right – Vasya Mar 28 at 16:19