# Can be infinitely equal to zero

## Limit values ​​x towards infinity

### What is the limit \$ x \$ towards infinity?

As part of a curve discussion, you have to draw the function graph of a function. More precisely: You draw a section of the function graph. Then the question still remains how the function behaves outside of this section. Which function values ​​are assumed when \$ x \$ is getting bigger or smaller?

Mathematically, this is expressed as follows:

• \$ \ lim \ limits_ {x \ to \ infty} ~ f (x) =? \$
• \$ \ lim \ limits_ {x \ to - \ infty} ~ f (x) =? \$

So it will be after Behavior in infinity asked the limit.

• The spelling “\$ \ lim \$” stands for “Limes”, Latin for “limit”.
• "\$ \ Lim \$" says what \$ x \$ should go against.

In the following we look at different methods for determining such a limit value.

### Calculate limit values ​​of functions through test applications

When determining the limit value by Test installation do the following.

You create a table of values. You choose values ​​for \$ x \$ that are getting larger (i.e. \$ x \ to \ infty \$) or smaller (i.e. \$ x \ to - \ infty \$). You calculate the associated function values ​​for these values. The behavior of these function values ​​then shows you what the function values ​​will eventually go against.

### example 1

Let's take a look at an example: \$ f (x) = \ frac {x ^ 2 + 1} {x ^ 2} \$.

Note that the Domain of definition this function is \$ \ mathbb {D} _f = \ mathbb {R} \ setminus \ {0 \} \$. This means that the function graph has (or can have!) A pole at \$ x = 0 \$.

You can see the corresponding function graph here. You can already see from this that the function graph clings to a straight line through \$ y = 1 \$ that is parallel to the \$ x \$ axis. The following applies:

\$ \ lim \ limits_ {x \ to \ infty} ~ f (x) = 1 \$

You can also determine the limit value for \$ x \ to- \ infty \$. This is also \$ 1 \$.

### Example 2

Let's look at another example: \$ f (x) = \ frac {x ^ 2-1} {x + 2} \$.

The domain of this function is \$ \ mathbb {D} _f = \ mathbb {R} \ setminus \ {- 2 \} \$. Here you can see the part of the function graph for \$ x> -2 \$. In the following table of values ​​you can see the function values ​​for some \$ x \$. You can see from the function graph as well as from the table of values ​​that the function values ​​keep getting bigger for ever larger \$ x \$. So the following applies:

\$ \ lim \ limits_ {x \ to \ infty} ~ f (x) = \$ "\$ \ infty \$"

In this case there is a improper limit value, i.e. not a finite number. That is why this is often written in quotation marks.

### Calculate limit values ​​of functions by simplifying terms

The procedure through Test installation is strictly not correct. Why? It could be by chance that you have found a sequence of \$ x \$ that approaches infinity, for which the corresponding limit value for the function comes out. In the case of a different sequence, the limit value could also be different. However, this is not the case for the functions under consideration.

The method is a little more “mathematical” Term simplification or Term rewriting.

For this we look again at the first example: \$ f (x) = \ frac {x ^ 2 + 1} {x ^ 2} \$.

The limit value is already known. This is \$ 1 \$.

The function term is now transformed. You can divide each summand in the numerator by the denominator and you get:

\$ f (x) = \ frac {x ^ 2 + 1} {x ^ 2} = 1 + \ frac1 {x ^ 2} \$

Now you can look at every single summand. You use the Limit sets.

The limit the Sum of two functions is equal to the sum of the limit values ​​of the individual summands. This also applies to differences:

\$ \ lim \ limits_ {x \ to \ infty} (f (x) \ pm g (x)) = \ lim \ limits_ {x \ to \ infty} f (x) \ pm \ lim \ limits_ {x \ to \ infty} g (x) \$

So:

\$ \ lim \ limits_ {x \ to \ infty} f (x) = \ lim \ limits_ {x \ to \ infty} \ left (1+ \ frac1 {x ^ 2} \ right) = \ lim \ limits_ {x \ to \ infty} 1+ \ lim \ limits_ {x \ to \ infty} \ frac1 {x ^ 2} \$

The left limit is \$ 1 \$, since a constant function is being considered, and the right limit is \$ 0 \$, so in total it follows:

\$ \ lim \ limits_ {x \ to \ infty} f (x) = \ lim \ limits_ {x \ to \ infty} 1+ \ lim \ limits_ {x \ to \ infty} \ frac1 {x ^ 2} = 1 + 0 = \$ 1

### Limits of completely rational functions

A completely rational function looks like this:

\$ f (x) = a_n \ cdot x ^ n + a_ {n + 1} \ cdot x ^ {n-1} + ... + a_2 \ cdot x ^ 2 + a_1 \ cdot x + a_0 \$

Where \$ n \$ is the Degree, the highest power, and \$ a_n \$ the corresponding coefficient of the rational function.

The Limit behavior of completely rational functions hangs

• on the one hand it depends on whether the degree \$ n \$ is even or odd and
• on the other hand, it depends on whether the coefficient \$ a_n \$ in front of the \$ x \$ with the highest power is positive or negative.

We will look at each of these using an example.

### Wholly rational functions with an even degree

The limit values ​​for \$ x \$ against plus and minus infinity of the function \$ f (x) = x ^ 2 \$ are to be determined. The function graph is a parabola that opens upwards. You can see here that both for always larger and for ever smaller \$ x \$, the function values ​​become larger and larger, that is, they approach infinity. You can of course do this Test installation check. So it applies

\$ \ lim \ limits_ {x \ to \ infty} ~ f (x) = \ lim \ limits_ {x \ to- \ infty} ~ f (x) = \$ "\$ \ infty \$".

If you look at the function \$ g (x) = - x ^ 2 \$ instead of \$ f (x) = x ^ 2 \$, you get a parabola that is mirrored on the \$ x \$ axis, i.e. that is open downwards. This applies

\$ \ lim \ limits_ {x \ to \ infty} ~ g (x) = \ lim \ limits_ {x \ to- \ infty} ~ g (x) = \$ "\$ - \ infty \$".

### Quite rational functions with an odd degree

For this we look at the function \$ f (x) = x ^ 3 \$ with the associated function graph. Here you can see the following limit values:

\$ \ lim \ limits_ {x \ to \ infty} ~ f (x) = \$ "\$ \ infty \$" and

\$ \ lim \ limits_ {x \ to- \ infty} ~ f (x) = \$ "\$ - \ infty \$".

Here, too, the mirroring on the \$ x \$ axis leads to a change in the sign of the limit values. For \$ g (x) = - x ^ 3 \$ applies

\$ \ lim \ limits_ {x \ to \ infty} ~ g (x) = \$ "\$ - \ infty \$" and

\$ \ lim \ limits_ {x \ to- \ infty} ~ g (x) = \$ "\$ \ infty \$".

### Summary

Depending on the degree \$ n \$, even or odd, and the corresponding coefficient \$ a_n \$, positive or negative, you can see the limit values ​​of a completely rational function

\$ f (x) = a_n \ cdot x ^ n + a_ {n + 1} \ cdot x ^ {n-1} + ... + a_2 \ cdot x ^ 2 + a_1 \ cdot x + a_0 \$

specify directly. The following table should give you an overview of this. 