One graph with three different types of asymptotes

Question

This graph is what I came up with while searching for one that had three different types of asymptotes, horizontal, vertical, and oblique.$f(x)=\sqrt{x^2+\frac{1}{x^2}}+x$

Can you provide other functions with three asymptotes? Please share your ideas.

@Ninad Munshi, For vertical an x in the denominator and for horizontal and oblique from both sides $\sqrt{x^2}+x$ — Henry, Dec 06 '22 at 19:51
Yes, we see your picture and the function you found. But you didn't ask any question about it. What is it that you want from us? Do you want confirmation that your function has three different types of asymptotes? Assuming that "horizontal, vertical, slanted" are considered different types for this, then yes, obviously it does. But we cannot give better than that "assuming" qualification, because we don't know the terminology of your book/class. So what is your question? — Paul Sinclair, Dec 07 '22 at 20:21
@Paul Sinclair, my question is clear. I am asking for other examples of functions with three different types of asymptotes. — Henry, Dec 08 '22 at 04:48
@David P, Your example has only vertical and horizontal asymptotes. Give an example with vertical, horizontal, and slant asymptotes. — Henry, Dec 08 '22 at 05:12
For more variety, you can also consider parabolic asymptotes and, more generally, polynomial (or even more generally, curvilinear) asymptotes. For the latter, see this paper [click "Download" on left side to get a .pdf of the paper]. — Dave L. Renfro, Dec 08 '22 at 08:33

score 1 · Answer 1 · answered Dec 08 '22 at 06:47

1

Another example is $\displaystyle f(x) = \frac{x-e^x}{1-e^x}$.

answered Dec 08 '22 at 06:47

Greg Martin

78,820

Your example is a nice one. – Henry Dec 08 '22 at 07:16

One graph with three different types of asymptotes

1 Answers1