# Can a function have asymptotes?

Feb 5, 2018

Yes, quite a variety...

#### Explanation:

Any of the following combinations are possible:

• No horizontal or slant asymptotes.

• One horizontal asymptote.

• One slant asymptote.

• Two horizontal asymptotes.

• Two slant asymptotes.

• One horizontal and one slant asymptote.

Any of the above can occur in combination with any one of the following:

• No vertical asymptotes.

• Any finite number of vertical asymptotes.

• A countable infinity of vertical asymptotes.

In addition, a function can have holes, i.e. removable discontinuities at points, spcifically:

• No holes.

• Any finite number of holes.

• A countable infinity of holes.

For example, we can construct a function with all of:

• One horizontal and one slant asymptote.

• A countable infinity of vertical asymptotes.

• A countable infinity of holes.

Define:

$f \left(x\right) = \sin \left(x\right) \csc \left(x\right) \tan \left(\frac{1}{x}\right) + \frac{x + \left\mid x \right\mid}{2}$

graph{sin(x) csc(x) tan(1/x) + (x + abs(x))/2 [-10, 10, -5, 5]}