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Asymptotes

Asymptotes are lines that a curve approaches but never touches or intersects. They provide insight into the behavior of the graph at extreme values.

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Asymptotes can be vertical, horizontal, or oblique.
For hyperbolas, asymptotes intersect at the center and define the slope of the branches.
The equations of asymptotes for a hyperbola centered at $(h, k)$ with horizontal transverse axis are $y = k \pm \frac{b}{a}(x - h)$. For a vertical transverse axis, they are $y = k \pm \frac{a}{b}(x - h)$.
Asymptotes help to sketch hyperbolas accurately by providing guidelines for their branches.
Asymptotic behavior describes how functions behave as inputs approach infinity or specific values.