Find all asymptotes of y=sqrt(x²-4)

Find all asymptotes of $y=\sqrt{x^2-4}$.

Step 1: Domain, no vertical asymptote

Defined for $|x|\ge2$; since $y\to0$ as $x\to\pm2$, there is no vertical asymptote.

Step 2: As $x\to+\infty$

\[ k = \lim_{x\to+\infty}\frac{\sqrt{x^2-4}}{x} = \lim_{x\to+\infty}\sqrt{1-\frac4{x^2}} = 1. \tag{1} \]

\[ b = \lim_{x\to+\infty}\!\left(\sqrt{x^2-4}-x\right) = \lim_{x\to+\infty}\frac{-4}{\sqrt{x^2-4}+x} = 0. \tag{2} \]

Step 3: As $x\to-\infty$

\[ k = \lim_{x\to-\infty}\frac{\sqrt{x^2-4}}{x} = -1, \qquad b = \lim_{x\to-\infty}\!\left(\sqrt{x^2-4}+x\right) = \lim_{x\to-\infty}\frac{-4}{\sqrt{x^2-4}-x} = 0. \tag{3} \]

Step 4: Verification

\[x=500\text{: } y-x\approx-0.00400\to0;\qquad x=-500\text{: } y+x\approx-0.00400\to0\ \checkmark\]