is it possible for a function to have both horizontal and oblique asymptotes ? please explain your reasoning

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degeneratecircle's profile pic

degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

If you want to get a little silly, you can use the example

`f(x)={(1 if x<=0),(1+x if x>0):}` , graphed below.

If your textbook uses the standard definition of "aymptote" that I've always seen, this is a perfectly good example. I bet your professor or TA would get a laugh out of seeing that.

mlehuzzah's profile pic

mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted on

 

You can have as many vertical asymptotes as you want (even an infinite number)

for example:

(This is the graph of  `"tan" ((pi x) / 2)`

But for horizontal and oblique asymptotes, you get at most 2 (total).  You can have one corresponding to `x-> -oo` and one corresponding to `x->oo`

You *can* have one of each.  An example might be:

 

This graph is `y=e^x +1` when `x<=0` and `y=(x^2+4)/(x+2)` when `x>0`

On the left hand side, you have a horizontal asymptote, `y=1`

On the right hand side, you have an oblique (slant) asymptote, `y=x-2`

 

So: the total number of horizontal and oblique asymptotoes must be 2 or fewer (you have a right hand side and a left hand side), but you can have one of each.

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