In the previous section we talked about comparing functions around infinity. In fact, this was just a special case, we can compare functions at any point of their domains. However, it is most useful at infinity, since at other points it is less intuitive. We will start by a brief survey of order from a theoretical point of view, then we will introduce asymptotes.

Comparing functions has been inspired by practical problems not unlike the intuitive evaluation of limits. In applications (above all in physics) it often helps if we replace a complicated expression by a simple one without making a large error.

Definition.

Letabe a real number, ∞, or−∞. Letf,gbe functions defined on some reduced neighborhood ofa.We say that

at f=O(g)aif there is a reduced neighborhoodUofaand a constantso that for all A> 0xfromUwe have

| f(x)| ≤A⋅g(x).We say that

at f≍ga, also denotedif there is a reduced neighborhood f= Θ(g),Uofaand constantsso that for all A_{1},A_{2}> 0xfromUwe have

A_{1}⋅g(x) ≤f(x) ≤A_{2}⋅g(x).We say that

at f=o(g)a, also denotedif f<<g,We say that

at f∼gaif

Sometimes we use the notation
*f* = *O*(*g*),*x*→*a**f* = *o*(*g*),*x*→*a*.

The second inequality can be expressed using just one constant, the condition
is equivalent to finding one positive *A* satisfying
*A*)⋅*g*(*x*) ≤ *f* (*x*) ≤ *A*⋅*g*(*x*).*A*)⋅*f* (*x*) ≤ *g*(*x*) ≤ *A*⋅*f* (*x*),

The limit condition in the definition of
*o*(*g*)

The first two definitions show rough comparison. Either *f* is
smaller than *g* up to a multiplicative constant (around *a*), or
it is about the same as *g* up to a multiplicative constant
*f* ≍ *g**a* the graph of *f* lies in a strip around the
graph of *g* whose width is determined by the two constants). The next
two definitions express a similar idea using limits, but they require more.
When *f* = *o*(*g*)*a*, it means that close
to *a* the function *f* is negligible compared to *g*. We will
return to this later. The condition *f* ∼ *g**a* these two functions
are pretty much equal.

Precisely and expressed mathematically, the latter two notions are stronger.
If *f* = *o*(*g*)*a*, then also
*f* = *O*(*g*)*a*. If
*f* ∼ *g**a*, then also
*f* ≍ *g**a*.

Terminology is a bit vague at this point. The two *o*-symbols are called
"big-O" and "little-O", for instance we would say that "*f* is a
little-*o* of *g* at *a*". These two symbols are also
sometimes referred to as "Landau symbols". These two comparisons are
orderings, that is, they have similar properties as the usual inequality. For
instance, they are transitive, that is, if at *a* we have
*f* = *o*(*g*)*g* = *o*(*h*),*f* = *o*(*h*);*O*".

The two relations
≍
and ∼ are equivalences, so they have similar property as the usual equality.
The relation
≍ and the big-O are tied
together similarly as inequality and equality. Namely, if at *a* we have
*f* = *O*(*g*)*g* = *O*( *f* ),*f* ≍ *g*.*o* and the ∼ relation.

Both pairs are connected in another way:
If *f* = *O*(*g*)*g* ≍ *h**a*, then
*f* = *O*(*h*) there.*f* = *O*(*g*)*f* ≍ *h**a*, then
*h* = *O*(*g*)

Similarly,
if *f* = *o*(*g*)*g* ∼ *h*,*f* = *o*(*h*).*f* = *o*(*g*)*f* ∼ *h*,*h* = *o*(*g*).

In the case of "big *O*" and both equivalences we often say that
*f* and *g* are of **the same order** (of
magnitude) at *a*. The equivalence
≍
is hard to prove directly, instead we usually use the following.

Fact.

If there is a real numbersuch that A> 0then

at f≍ga.

We already mentioned the usefulness of the relation ∼ when guessing limits. This comes also very handy in physics. In particular one often compares functions to powers. If we have a more complicated function and we want to guess its behavior near some point, we can try to compare different parts of this function to powers and then we know which parts can be ignored.

**Example:** Assume that we have a function
*f* + *g**f* ∼ *x*^{A}*g* ∼ *x*^{B}*A*,*B* > 0.*A* > *B*,*g* around infinity, mathematically,
*f* + *g*) ∼ *f**A* > *B**g* = *o*( *f* )

In general we have this:

Fact.

Ifat g=O(f)a, then( atf+g) ≍fa.

Ifat g=o(f)a, then( atf+g) ∼fa.

In fact, we have been using these considerations in the previous section on intuitive evaluation at infinity. In physics and numerical mathematics they also often do comparison at zero. There it is a bit tricky, since the scale of powers works differently. In particular, larger powers are actually beaten by smaller powers at 0.

**Example:** Assume that we have a function
*f* + *g**f* ∼ *x*^{A}*g* ∼ *x*^{B}*A*, *B* > 0.*A* < *B*,*g* around 0, mathematically,
*f* + *g*) ∼ *f*

**Example:** The function
*f* (*x*) = *x*^{2} − *x**x*^{2} at infinity; that is,
*f* ∼ *x*^{2}*f* (*x*) = *x*^{2} − *x**x* at 0 (check!), that is,
*f* ∼ *x*

The little-*o* comparisons are also reversed. On the one hand, at
infinity we have
*x* << *x*^{2},*x* = *o*(*x*^{2}).*x*^{2} << *x*,*x*^{2} = *o*(*x*).

Physicists, engineers and numerical mathematicians
use such stuff quite often. They would determine the order of the
whole expression and then ignore all its terms that are *o* of this
order.

One can also make comparisons at *a* from one side only. Definitions and
properties are analogous.

**Example:** The function
*f* (*x*) = *x*^{2} − ln(*x*)*x*^{2} at infinity, that is,
*f* ∼ *x*^{2}*x*) << *x*^{2}*x*) = *o*(*x*^{2})

On the other hand,
*f* (*x*) = *x*^{2} − ln(*x*)*x*)*f* ∼ ln(*x*)*x*^{2} << ln(*x*)*x*^{2} = *o*(ln(*x*))*f* at 0 from the right, but that's life. Indeed, check that
there is no *A* for which we would have
*f* ∼ *x*^{A}

Before we get to asymptotes, we will tie up the concepts from this section to
the concepts of the previous section. There we had two notions. Order allowed
us to guess limits and is the same as the order ∼ here. It is rather precise, for instance
*x*^{2} + *x**x*^{2}*x*^{2} + *x**x*^{2} at
infinity. This notion is similar to the similarity
≍, but it is somewhat more
loose, since it does not consider the sign. For instance,
*x*^{2}*x*^{2}, but we don't have
*x*^{2} ≍ *x*^{2}.

We had several notions comparing functions, but none of them help us draw
graphs. For that we need a different notion. Note that even if we use the
strongest notion we have above, it is still not enough to draw the graph
properly. Indeed, for instance
*x*^{2} − *x* ∼ *x*^{2}*x* and therefore the graphs
are spreading apart as we go to infinity.

We will therefore ask a different question: What is the difference between
two functions near a certain point *a*?

Definition.

Letabe a real number, ∞, or−∞. Letf,gbe functions defined on some reduced neighborhood ofa.We say that the graph of

fis asymptotic for the graph ofgataif

Note that this relation is symmetric, that is, if *f* is asymptotic for
*g* at *a* (we say it like this for short), then also *g* is
asymptotic for *f* at *a*. In fact, this relation is an
equivalence.

What is the relationship between this notion and the order we covered above?
There is one special case. If *f* and *g* are not separated from 0
as we approach *a* (see below), then these two notions are independent.
Otherwise asymptoticity is stronger.

Fact.

Assume that there is a reduced neighborhoodUof a pointaand a constantsuch that k> 0| andf| >k| ong| >kU. If these two functions are mutually asymptotic ata, then alsof∼g.

However, sometimes these notions are the same.

Fact.

Assume that functionsf, resp.ghave non-zero limitsA, respBata. Then the following are equivalent:

(i)A=B;

(ii)fandgare mutually asymptotic ata;

(iii)at f∼ga.

Otherwise asymptoticity usually brings more information. The most typical
case is when the functions go to (minus) infinity at (minus) infinity, then
asymptoticity is strictly stronger than the notion of order. For
instance, *x* ∼ (*x* + 7)

One reason why we almost exclusively use asymptoticity with *a* equal to
infinity or minus infinity is that it does not help us at all when drawing
graphs around proper points. Consider some particular *a*, for instance
*a* = 0.*x*^{2} and
*x*⋅sin(1/*x*)*a*, so when we subtract them, we get the limit as
zero again, therefore they are mutually asymptotic at 0. However, when you
compare their graphs, you will see that their shapes are totally different
around the origin.

Situation is therefore usually as follows. We are given a function and we know that at infinity it goes to infinity (or negative infinity etc.). We want to know something more about the manner in which it goes to that infinity. We would like to compare it to some other, nicer function that we already know. Thus we would look for a candidate for an asymptote. Since asymptoticity is very strong, we seldom find such a candidate. But if we do, we can make a very good guess concerning the graph of the given function around infinity (resp. negative infinity).

Most often we look for straight lines as asymptotes. Once we decide that we are interested only in straight lines as asymptotes, we may extend the notion a bit further and we also get a nice algorithm for determining asymptotes (if any exist). Thus the focus shifts away from the notion of order and we prefer to leave the topic of straight asymptotes to a different section, asymptotes in Derivative - Theory - Graphing functions.