Can a function have asymptotes?

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:

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

Yes, a function can have asymptotes. Asymptotes are lines that a graph approaches but never intersects as the independent variable approaches infinity or negative infinity. Asymptotes can be vertical, horizontal, or slant (also called oblique). Vertical asymptotes occur when the function approaches a vertical line as the independent variable approaches a certain value. Horizontal asymptotes occur when the function approaches a constant value as the independent variable approaches positive or negative infinity. Slant asymptotes occur when the function approaches a linear function as the independent variable approaches infinity or negative infinity.