What are the asymptotes of the hyperbola? Can you figure out how to generalize the asymptotes in terms of $ a$ and $ b$?

Use slope fields and the equation of graphs to find their asymptotes. Can you find the diagonal asymptote of the graph $ \frac{x^2+1}{x}$?

Please watch the video below to better your understanding of Asymptotes.

- Asymptote – a line that approaches infinitely close to a given curve.
- Vertical Asymptote – the asymptote that exists when x approaches infinity at a given point
- Horizontal Asymptote – horizontal line that represents the value a function approaches as x approaches infinity

**Text to follow along with video:**

If we were going to show the graph of the hyperbola starting at some arbitrary point and follow the slope lines as you can see we’d keep getting closer to the diagonal line. If we were to start exactly on this diagonal line our graph would end up being just both of our diagonals.

In this way the slope field is divided into four sections bounded by these diagonals. These diagonal lines are what we call asymptotes. Asymptotes are imaginary lines on a graph that a solution to a differential equation approaches infinitely close to.

As you have seen when working with hyperbolas they approach these diagonal asymptotes and as the function goes off into infinity the graph becomes closer and closer to these lines.

Hyperbolas aren’t the only graphs with asymptotes though. Take the equation $ \frac{1}{x}$ you’ve most likely seen this graph before and as you can see there is some interesting action as the graph gets further and further from the origin. There are two asymptotes for this graph. One is along the $ x$ axis as $ x$ approaches infinity, the second is right here on the $ y$ axis, as $ x$ gets closer and closer to zero. As seen from these examples asymptotes can have any slope. They just need to attract the graph of a function closer and closer to them, however it is worth noting that unless you start your graph on the asymptote, it will never actually equal the asymptote lines. It will approximate that line better and better as you get closer to infinity, but it won’t ever be equal.

So far we’ve seen these asymptotes as something being approached and never crossed, however there are functions that oscillate around their asymptote. If we take the graph of $ \frac{\sin{x}}{x}$ and graph it we start with what looks like a regular sine graph, however as we get further and further out the oscillations of the graph become less and less pronounced. In fact if we were to take the limit of this graph as $ x$ approaches infinity we would see that the graph approaches zero. So even though there is oscillation the graph still trends towards a specific line.

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