# What is 1 divided by infinity

### content

Average rating: 3.21 of 5 with 24 votes.

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.