**Q: Hi Joel, could you run through the basics of asymptotes?**

The fundamental idea of asymptotes is that it is a limit. For example, consider the function . What does it tend to as gets really, *really* large? The answer, of course, is , since divided by a reaaaaaaaally huge number gives us a reaaaaaaaally small number, which approaches . We can graph this function this way:

Since tends toward , we call our *horizontal* asymptote.

What about the ? How do we make *that* go to infinity? We do this by considering divided by a reaaaaaaaally small number. At the threshold, we take divided by and get an undefined number. So to get to that extreme, we let the denominator equal . In this case, , which just so happens to be our vertical asymptote.

BUT, what about this function, , which is the initial function but translated by 2 units in the positive *–*direction? We ask the same questions: what happens when gets really, *really* large and how can we make really, *really* large?

When gets really, *really* large, it follows that gets really, *really* large and equals divided by a reaaaaaaaally huge number, which gives us a reaaaaaaaally small number, which approaches . Thus the horizontal asymptote *remains* as .

How can we make go to infinity? We do this by letting the denominator equal . In this case, and . Hence, that is our asymptote.

BUT, what about this function, , which is the second function but translated by 3 units in the positive *–*direction? We ask the same questions: what happens when gets really, *really* large and how can we make really, *really* large?

When gets really, *really* large, it follows that gets really, *really* large and equals divided by a reaaaaaaaally huge number, which gives us a reaaaaaaaally small number, which approaches . Thus, *overall*, tends towards since the tends toward . Therefore is our horizontal asymptote.

How can we make go to infinity? We do this by letting the denominator equal . In this case, and . Hence, that is our asymptote.

BUT, what about this function, , which is the third function *plus *??? We ask the same questions: what happens when gets really, *really* large and how can we make really, *really* large?

When gets really, *really* large, it follows that gets really, *really* large and equals divided by a reaaaaaaaally huge number, which gives us a reaaaaaaaally small number, which approaches . Thus, *overall*, tends towards since the tends toward . Therefore is our asymptote. It’s not horizontal, though. We call these *slanted* asymptotes **oblique**.

How can we make go to infinity? We do this by letting the denominator equal . In this case, and . Hence, that is our vertical asymptote.

BUT, what about this function, , which is the generalised third function? Bonus marks if you can describe the sequence of transformations from the third function to this. When gets really, *really* large, it follows that gets really, *really* large and equals divided by a reaaaaaaaally huge number, which gives us a reaaaaaaaally small number, which approaches . Thus, *overall*, tends towards since the tends toward . Therefore is our oblique asymptote.

How can we make go to infinity? We do this by letting the denominator equal . In this case, and . Hence, that is our vertical asymptote.

In sum, to find the asymptotes, we ask the two questions

- What happens when gets really,
*really*large? (to obtain horizontal or oblique asymptotes) - What happens when the denominator equals zero? (to obtain vertical asymptotes)

**Footnote*: The last function assumes as division by zero is undefined. Also, if , we actually get a linear function , now assuming that as division by zero is undefined. Note also that if then we get the initial examples without oblique asyptotes.