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Calculate limit values ​​of functions

Here the determination and calculation of the limit values ​​of functions is simply explained. Here is an overview of the page (click on a topic to scroll there directly):

  1. The Limes in general
  2. Limit values ​​towards infinity simply explained
  3. Limits declared against a finite number (e.g. 0)
  4. Calculate limit values
  5. Limit value calculation rules

Limit values ​​can be specified with the Limes. The limit describes what happens when you insert values ​​for a variable that are getting closer and closer to a certain value. Under the "lim" is the variable and which number it goes against (ie which value the variable is getting closer and closer to). After the "lim" there is the function in which the values ​​for x are inserted, for example:

This notation means that you insert values ​​for x in the function 1 / x, getting closer and closer to infinity. You can't use an infinite value, but with the Limes you can "see" what would come out as infinite. One then speaks “Limes against infinity”. Of course, this also works with all other values, not just for infinity.

Limit values ​​at infinity

Limit values ​​in infinity describe what happens to the function, i.e. what value the function approaches more and more when x approaches infinity (that is, when x becomes larger and larger to infinity). Here x can run towards + and - infinitely, i.e. it can become smaller or larger. It then looks like this in mathematical notation:

The limit value then looks graphically as shown here for x ^ 2. If you want to have the limit for + ∞ or -∞, look what the function "does in that direction". Here it goes towards infinity in both directions.

Limit values ​​in the finite are values ​​that the function assumes when it approaches a certain value. This is often used on definition gaps to check what is going on near them. You can approach the value from the left or right, i.e. approach the definition gap from the negative side, or from the positive side, because sometimes different limit values ​​result. This is then noted as follows:

Left is the approach to zero from the positive side and right from the negative. Drawn it looks like this:

Graphically the whole thing looks like this (for 1 / x). So you look where the function is "going" when you approach a number from the positive side and once from the negative. As you can see, this gives 2 different results.

In order to determine a limit value, one has to consider what happens to the function if one uses values ​​that are closer and closer to the examined value, i.e. the value against which the x runs.

Procedure for limit values ​​towards infinity

  • Look where the x is, e.g. in the exponent, denominator, base ... and see what happens when x gets bigger / smaller.
  • If there are several x, look at the x that is growing the most, i.e. the one that has the most influence on the limit value. E.g. the x with a higher exponent has more influence than the one with a smaller one. Here is a small ranking list, if several x occur in a function, from the smallest influence to the largest (1st smallest influence, 4th largest influence):
    1. Root of x
    2. x without exponent (or exponent 1)
    3. x with the highest exponent
    4. x is itself in the exponent You then only have to look what happens to the most influential x for infinity, that is then the limit value.
  • Just exclude the highest power, because wherever the power is in the denominator, it becomes 0 and so you can quickly see what comes out.


Procedure for limit values ​​against fixed values

Here it is explained using the example of x against 0:

  1. Put in zero for every x and see what comes out, this is sometimes already the limit value.
  2. But if you have a 0 in the denominator (which you are not allowed to do), it approaches infinity, because the denominator gets smaller and smaller the closer the value comes to zero.
  3. But if you have a 0 in the numerator and denominator, if you insert x = 0, it depends on whether the numerator or denominator degree is greater, or where the x with the greater influence is, it "wins", i.e. if If the numerator degree is larger, it goes towards 0 and if the denominator degree is larger towards infinity. However, if the numerator and denominator degree are also the same, then the limit value is the quotient of both factors in front of the x with the highest exponent in the numerator and denominator.


Limits for certain functions:

The limit of a power function is given by:

  • + ∞ for n> 0
  • 1 for n = 0
  • 0 for n <0
  • + ∞ for even n> 0
  • -∞ for odd n> 0
  • 1 for n = 0
  • 0 for n <0

The limit of the exponential functions is given by:

  • + ∞ for a> 1
  • 0 for a between 0 and 1
  • - there is none for a less than 0
  • 0 for a> 1
  • + ∞ for a between 0 and 1
  • - for a <0 there is none

Fractional rational functions

In the case of fractional-rational functions, the highest exponent in the numerator (n) and denominator (m) is important, but also the factors before the highest power in the numerator (a) and denominator (b).

The following applies:

  • n is the highest exponent in the numerator
  • m is the highest exponent in the denominator
  • 0 for n
  • an÷ bm for n = m

  • + ∞ for n> m and an÷ bm >0