Proof of the asymptotes

Why are the lines $y=\pm\frac{b}{a}x$ asymptotes of the hyperbola? The idea is to compare the $y$-coordinate of the curve with that of the line when $x$ becomes very large.

Proof — The asymptotes

From the canonical form, isolating $y$: $y = \pm\,b\sqrt{\tfrac{x^2}{a^2} - 1} = \pm\,\tfrac{b}{a}\sqrt{x^2 - a^2}.$ For $|x|\to\infty$ one has $\sqrt{x^2-a^2} = |x|\sqrt{1 - a^2/x^2}\to |x|,$ so $y \approx \pm \tfrac{b}{a}|x|$, that is $y \approx \pm\tfrac{b}{a}x$ (one sign or the other depending on the branch). The lines $y = \pm\tfrac{b}{a}x$ are therefore the asymptotes. ∎

In practice, for large values of $|x|$ the term $a^2/x^2$ becomes negligible and the root behaves like $|x|$: the difference between the curve and the line tends to zero.

Links

Topics: Homographic hyperbola
Concepts: Asymptote · Hyperbola
Functions: Hyperbola
Skills: Limiting-case analysis · Proving