What is 1 divided by infinity

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We now consider a broken rational function and the boundaries of the domain. You will learn how to transform function terms and calculate limit values.

Status: 03/18/2013 | archive

Now we have already determined two clear limit values. In general it can be said:

General statement on the limit value

If the denominator of a function term with a constant numerator approaches zero, the limit value is infinitely large. If the denominator approaches infinity, the limit is zero.

The boundaries of the domain

Calculation at the edges of the definition range - please click on the magnifying glass.

Knowing this, we will now look more closely at a broken rational function. In order to get an idea of ​​the course of the graph, we will examine the function at the edges of the domain. Please take a look at the adjacent function. The function is not defined at the point x is equal to zero. It therefore has a definition gap at this point. The definition set is therefore R without zero.

The function is defined from minus infinity to zero - but no longer exactly zero - and from zero or a little more than zero to plus infinity. We now examine the behavior at the edges. Let's start for x towards plus infinity. It gives infinite minus 1 by infinite squared. We already have an unclear limit value. Because infinite minus 1 is infinite and in the denominator infinity to the square is also infinite. We do not know what is infinite through infinity. We only know that a constant number goes to zero through something infinitely large and that a constant number results in something infinitely large through something very small, i.e. zero.

So we have to try to transform the function term in such a way that corresponding unambiguous sub-terms arise.

Transformation of the functional term: Please click on the magnifying glass.

You can see the forming on the right - please click on the magnifying glass. After factoring out and shortening x, what remains is: Limes from x to plus infinity from 1 minus 1 through x in the numerator through x in the denominator.

Calculation of the first boundary value

A clear sub-term

Now we have a unique sub-term with 1 through x. If x becomes infinitely large here, the limit value of 1 goes through x to zero. All that remains is: Limes x towards infinity from 1 through x. And we already know that: This limit value is zero. We therefore know the first marginal point.

Calculation of the second boundary value

The next edge point - if we move from plus infinity to minus infinity on the number line - is zero. How does our function term behave here?

Calculation of the second boundary value

If we substitute zero for x, we get minus 1 through zero. With 1 through x for x towards zero, we had the definite limit plus infinity. For minus 1 times 1 by x times 1 by x, this means that infinity is multiplied by minus 1, and leads to the limit value minus infinity.

Remaining marginal value calculation

Calculation of the third marginal value: Please click on the magnifying glass.

The second boundary value is therefore also clear. Basically the third one too. If we approach the x-value zero, coming from minus infinity, the marginal value also results in minus infinity. Thus only the boundary value for x towards minus infinity is missing.

Remaining marginal value calculation

Exactly as with the consideration for x against plus infinity we exclude x in the numerator and get limits x against minus infinity from x times parentheses to 1 minus 1 through x parentheses closed in the numerator, through x times x in the denominator.

Remaining marginal value calculation

A shortened x leads to limit x towards minus infinity from 1 minus 1 through x through x. 1 through x is a clear limit value at x towards infinity, namely zero.
And again a clear limit value with zero for the entire functional term.

Limit value calculation and graphical representation - please click on the magnifying glass.

Now we have calculated all the boundary values. Using a table of values, we will now draw the graph of the function and check whether the limit value determinations are correct. To do this, we insert the x values ​​from the table of values ​​into the function term: x equals minus 4 results in the function value minus 5 sixteenths, x minus 3 results in minus 4 ninths, and so on. At the point x equals zero we have a definition gap. (Please click on the picture opposite). The curve shown here is shown graphically. For x towards plus infinity and x towards minus infinity, the graph approaches the x-axis, i.e. the function value 0. For x towards zero, the graph approaches the function value minus infinity from both sides of the f (x) axis.

General statement on the limit value

If the denominator of a function term with a constant numerator approaches zero, the limit value is infinitely large. If the denominator approaches infinity, the limit is zero.

Conclusion

  • Only clear limit values, such as a numerical value through x, are always applicable. Here the limit value can be clearly determined for values ​​towards plus or minus infinite as well as towards zero.
  • If zero through zero or infinity through infinity occurs in the fractional term, the limit values ​​are unclear. However, by cleverly decomposing numerator polynomials and denominator polynomials, often also by simply factoring out common zeros, common zeros can be found and reduced. A clear limit value thus arises from an as yet unclear limit value.