We recognize three kinds of asymptotes: vertical, horizontal, and oblique. Note that formally, horizontal asymptotes actually belong among oblique ones, but for practical purposes it is better to treat them as a separate case.

Consider a function *f*.

**Algorithm for vertical asymptotes:**

**Step 1.** Identify points where the function might have vertical
asymptotes. Suspicious points are: endpoints of intervals from the
definition; endpoints of intervals of the domain; points where continuity is
suspect.

**Step 2.** For each suspicious point *a*, find all one-sided limits
at *a* that make sense (depends on which side of *a* the function
is defined). If at least one of these limits is improper, then the function
has a vertical asymptote at *a* (or put another way, the line
*x* = *a*

Note: If we try one one-sided limit and it is improper, we already have a
vertical asymptote there and there is no need to calculate the second
one-sided limit (if it makes sense at all).

**Algorithm for horizontal asymptotes:**

**At infinity:** If the function is defined on some neighborhood of
infinity, find its limit at infinity. If this limit is proper and equal to
*L*, then the line *y* = *L**f* at infinity.

**At negative infinity:** If the function is defined on some neighborhood
of negative infinity, find its limit at negative infinity. If this limit is
proper and equal to *L*, then the line
*y* = *L**f* at negative infinity.

**Algorithm for oblique (non-horizontal) asymptotes:**

**At infinity:**

**Step 1.** If the function is defined on some neighborhood of
infinity, find its limit at infinity. If this limit is proper or does not
exist, we do not have an oblique asymptote at infinity. If it is improper, go
to the next step.

**Step 2.** Find

If this limit diverges (or is zero), then *f* does
not have an oblique asymptote at infinity. If it converges (and is not zero), go
to the next step.

**Step 3.** Find

If this limit diverges, then *f* does
not have an oblique asymptote at infinity. If it converges, then the line
*y* = *A**x* + *B**f* at infinity.

**At negative infinity:**

**Step 1.** If the function is defined on some neighborhood of negative
infinity, find its limit at negative infinity. If this limit is proper or
does not exist, we do not have an oblique asymptote at negative infinity. If
it is improper, go to the next step.

**Step 2.** Find

If this limit diverges (or is zero), then *f* does
not have an oblique asymptote at negative infinity. If it converges (and
is not zero), go to the next step.

**Step 3.** Find

If this limit diverges, then *f* does
not have an oblique asymptote at negative infinity. If it converges, then
the line
*y* = *A**x* + *B**f* at negative infinity.

**Example:** Investigate asymptotes of

**Solution:** *D*( *f* ) = (−∞,−1) ∪ (−1,∞),

Candidates for vertical asymptotes: the proper endpoint
*x* = −1

We do not have to evaluate the second one-sided limit, we already know from
this limit that *f* has a vertical asymptote at −1.

Asymptote at infinity: We check on the limit

This shows that there is no horizontal asymptote at infinity. However, there is a chance for an oblique one. We follow the algorithm.

There's still a chance.

We conclude that the line
*y* = *x*/2 + 1

Asymptote at negative infinity: We check on the limit

This shows that there is no horizontal asymptote at negative infinity. However, there is a chance for an oblique one. We follow the algorithm.

There's still a chance.

We conclude that the line
*y* = *x*/2 + 1

By the way, if you want to see how this function looks like, check out the Example in Methods Survey - Graphing - Overview).

For more insight and another example see Asymptotes in Theory - Graphing, see also appropriate problems in Solved Problems.