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.