Can a function have asymptotes?

1 Answer
Feb 5, 2018

Answer:

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(x) = sin(x) csc(x) tan(1/x) + (x + abs(x))/2#

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