Definition 4.8.1
Let

be a function defined on a neighborhood of

and let

be the line with equation

. The line

is an asymptote to the graph of

if
![$ \underset{x \rightarrow +\infty}{\lim} [f(x)-(ax+b)]=0$](img723.png)
.
There is a similar definition for a neighborhood of
. Please write it down.
Example 4.8.2
Let

. As
![$ \underset{x \rightarrow +\infty}{\lim} [f(x)-2x]=
\underset{x \rightarrow +\infty}{\lim}(\frac 1x)=0 $](img725.png)
,
the graph of

has an oblique asymptote whose equation is

.
You can check that the
axis is a vertical asymptote to this graph, according to def vert asympt.
Figure 13:
A curve with an oblique asymptote.
 |
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.
Example 4.8.3
Take

.
On Figure 14, we display the graph of
.
Figure:
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.
Example 4.8.4
Take

We compute the limit of

when

tends to infinity.
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
.
Figure:
The graph of
.
 |
Example 4.8.5
Let

. As the sine function is bounded over

, the limit of

at

is

. Let us look for an oblique asymptote:
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.
Figure:
The graph of
.
 |
Noah Dana-Picard
2007-12-28