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Let A be the hyperbola with the equation $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ is the $x$-intercept and $b$ is the $y$-intercept.

Given this it can be calculated that the lines $\displaystyle y=\frac{b}{a}\cdot x$ and $\displaystyle y=-\frac{b}{a}\cdot x$ are both asymptotes to the hyperbola.

The question is simply: are these asymptotes significant? And if so, why?

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    $\begingroup$ These asymptotes are significant because they are asymptotes: see J.G.'s answer. Where you looking for something more substantial than that? $\endgroup$ – Mike Pierce Aug 13 '17 at 23:03
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    $\begingroup$ Did you look at Wikipedia article on hyperbola's asymptotes? What does "significant" mean exactly? $\endgroup$ – Conifold Aug 13 '17 at 23:05
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    $\begingroup$ In short: lines are easier to study than conics, and at a sufficiently far distance from the center, hyperbolas almost look like their asymptotes. $\endgroup$ – J. M. is a poor mathematician Aug 14 '17 at 4:58
  • $\begingroup$ Just to add a different concept to this thread: in school, we would plot hyperbolas by drawing a box that corresponded to the axes; then we would draw the asymptote through the diagonals of this box. When it came time to plot our hyperbola, we didn't plot any points; rather, we simply guesstimated a curve around these asymptotes. So in reality we were graphing asymptotes. I'm sure you can see how this fits in with the answers given. $\endgroup$ – gen-z ready to perish Aug 14 '17 at 8:38
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The hyperbola given by the equation

$$ x^2 - y^2 = 1$$

looks like this:

Zoomed out picture of a hyperbola (image source: wolfram alpha)

At this scale, the hyperbola is virtually indistinguishable from the union of its asymptotes. Since lines are easier to understand than hyperbolas, at this scale it's much easier to understand the hyperbola in terms of its asymptotes.

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Here is a graph of

$$\frac{x^2}{4}-\frac{y^2}{9}=1$$

at a large scale.

I would also include the graphs of the asymptotes

$$ y=\pm \frac{3}{2}x$$

but I think the result would be obvious.

graph of hyperbola

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Yes, the asymptotes tell you the global behavior of the hyperbola, and in general it is extremely useful to be able to explain the global behavior of something complicated using simpler things that you already understand well (such as linear functions).

Specifically, they tell you that a hyperbola becomes linear far away from its center.

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For sufficiently large $x$ the hyperbola is arbitrarily close to its asymptotes.

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  • $\begingroup$ That's the definition of a limit, and hence of an asymptote. But I agree it's important to keep that in mind to understand the usefulness of it. $\endgroup$ – someonewithpc Aug 14 '17 at 14:16

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