There is a similar definition for a neighborhood of . Please write it down.

You can check that the axis is a vertical asymptote to this graph, according to def vert asympt.

We will now see an algorithm to check whether the graph of a given function has an oblique asymptote or not. WLOG, we suppose that .

- We compute
.
- If it is infinite, there is no oblique asymptote, and we are done.
- Otherwise, we denote by this limit and proceed to the next step.

- We compute now
.
- If it is infinite, there is no oblique asymptote, and we are done.
- Otherwise, we denote by this limit and the line whose equation is is a an asymptote to the graph of the function.

On Figure 14, we display the graph of .

We work according to the algorithm above. Suppose that ; then:

thus

Now,

whence

We conclude that the line whose equation is is an asymptote to the graph of .

Working in a similar way for , we can show that the line whose equation is is an asymptote to the graph of . This result is also a consequence of the fact that is an even function, therefore its graph is symmetric about the axis.

Therefore the graph of the function has no oblique asymptote for arbitrarily close to infinity. A similar computation shows that the same situation occurs in a neighborhood of .

On Figure 15, we display the graph of .

By the sandwich theorem (6.1), we have:

thus

If there is an oblique asymptote, its slope must be equal to 1. Now:

and this has no limit at infinity. Similar work has to be done at . Hence, the graph of has no oblique asymptote.

Noah Dana-Picard 2007-12-28