Математика. Материалы курса

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Something is in the air. These are uncertainties, but we will put a couple of them in order today. We are going to consider the uncertainties 0/0 and infinity divided by infinity and we’ll see how they are solved.

The first uncertainty of these is infinity divided by infinity.

What should we remember? We should keep this rule in mind.

If we calculate the limit of the ratio of two polynomials with x tending to infinity, then the rule is: we must divide the numerator and denominator of the fraction by the highest power of the variable x.

Let's consider an example of how this rule works. Here is an example: the limit of the ratio of polynomials for x tending to infinity.

 This is an uncertainty: infinity divided by infinity.

 How do we find it out?

We plug infinity instead of x and understand that there is some infinitely large value in both the numerator and denominator.

 The highest power of the variable in this fraction is x squared.

We divide the numerator and denominator by the x-squared. Look at what we get. We get a fraction.

If we plug infinity instead of x, then there is no uncertainty. We remember that a number divided by infinity is 0 in the theory of limits.

 As a result, the answer is instant: the uncertainty has disappeared; we have solved it.

 Example 2. The change is not great.

In example 2, I removed the power, we had 3 x square, now we have 3x. The power is one less. The power of the numerator is one less than the power of the denominator. Still the highest power of the variable that occurs here is x-squared.

 The transformations are identical every time. We divide it by x-squared and what do we get? We do not have any uncertainties anymore; the numerator turned out to be 0, the denominator is 5, 0 divided by 5, it is an allowed action, the answer is 0.

Example 3. The power of the numerator is greater than the power of the denominator, but still the highest power is x squared. We divide it, and the uncertainties have disappeared. When calculating the limit of the numerator and denominator, we get 3 in the numerator and 0 in the denominator. In the theory of limits, this is a permitted action. We get infinity as the answer. All problems are solved.

 What can you notice? Every time I circle the answers we have received. The answer is 3 / 5. How does it relate to example 1? These are the coefficients for x-squared, for the highest power of the variable, 3 and 5.

 What are 0 and 5 in example 2? These are the coefficients for the highest power of the variable, for x squared, the numerator is 0, and the denominator is 5.

What are 3 and 0 in example 3? The fraction obtained at the last stage. These are the coefficients for the highest power of the variable x squared. 3 and 0.

Therefore, in such examples, if we are not required to perform identical transformations, we can get answers very quickly, almost viva voce.

So let's make a conclusion which sometimes will be helpful: the limit of the ratio of two polynomials at infinity, i.e. as x approaches infinity, is equal to the ratio of the coefficients for the highest power of the variable of these polynomials.

A very useful rule.

 So, let's consider another uncertainty – 0 by 0. How can we deal with it? Note that each uncertainty has its own rule.

You can check that this uncertainty is not solved by the transformation that we have just considered, and nothing will work.

What should we remember? 0 divided by 0.

It turns out (it is not said here), with the relation of the polynomial and so what should be done?

You need to factor the numerator and denominator.

Note, you will definitely then reduce by x minus a.

At this point, the specified uncertainty will disappear. x tends to a, and we will reduce by x minus a.

Let's take some examples.

First, let us discuss how to expand numbers into factors.

We can use the root formula for a square trinomial and factor it, or we can use the Vieta theorems.

We need to remember how the roots are related to the coefficients of the square trinomial. Bézout’s theorem is another important theorem. What is it about? If a real number is the root of a polynomial, then this polynomial is factored into multipliers, one of which is x minus a. We are going to use it while reducing a polynomial.

 Let's take some examples.

 This is an essential rule: we calculate any limit, plugging 1 instead of x here; then we calculate, and get 0 divided by 0.

 I do not show how we have done it (we calculate the discriminant, find the roots of the square trinomial, and write the coefficients 3 and 5).

 x tends to 1, and the numerator and denominator have just the necessary multipliers x minus 1.

If you do not get them, then look for an error.

You might make a mistake in factoring, or there was no uncertainty.  

According to Bézout’s theorem we are to get a multiplier of x minus 1. Now we have to reduce it, and then plug 1 instead of x. We open the brackets, and the uncertainty is gone.

 We get the answer.

 Let's take another example.

We have the ratio of polynomials. We plug 2 instead of x and get 0 in the numerator and 0 in the denominator. This is an uncertainty, in other words, we don't know what the answer will turn out to be.

Note that the uncertainty will disappear if we reduce it by x minus 2.

These are not square trinomials. How are we going to factor it?

 There are different techniques. We can factor it using some identical transformations or Bézout’s theorems.

 Since 2 is the root of both the numerator and the denominator, then both the numerator and the denominator are factored into multipliers, one of which is x minus 2.

When we reduce it by x minus 2, what is left? The division of the numerator and denominator by x minus 2 can be done in a column. You can try to do it yourself.

 I strongly recommend that you complete this task yourself.

 We divide the numerator by x minus 2 according to the same rules we use with numbers.

We divide the denominator by x minus 2. I will now show the result of this division.

Try to understand how these actions are carried out.

Note that the remainder must have zeros, since according to Bézout’s theorem division is carried out without a remainder. In the quotient, we get polynomials of the second and third power – these are the missing multipliers in factoring.

 Thus, we get x is minus 2 and the second multiplier in the numerator and denominator.

This is the result of dividing the polynomials by x minus 2.

 We reduce it by x minus 2, plug 2 instead of x and note that the uncertainty has disappeared. Sometimes it happens that there is still 0 by 0. You need to divide again, or factor it into multipliers in some other way.

 Now let us consider factorization followed by reduction by x minus a, which is what x tends to do.

The situation is slightly different here. Example 3. If we plug 1 instead of x, we get 0 by 0. These are not polynomials, and we can't factor them.

However, the rule is the same: the uncertainty disappears after we reduce it by x minus 1.

 Where do we get x -1? We need to restore the polynomials. What are we going to use? If we multiply the numerator by the sum the root of x + 1, then we get just x minus 1, this is the product of a minus b by a plus b (the formula for the difference of squares).

For the denominator (there is a root of the third power), we need to use the formula for the difference of cubes.

 If we multiply the third-power root of x minus 1 by the incomplete square of the sum, we get the required expression x minus 1.

 Thus, first, we write down the fraction, and use the first formula.

 The transformations are identical, that’s why we multiply the numerator and denominator by the root of x + 1. This is what we do for the numerator.

In order to convert the denominator, we multiply it by the incomplete square of the sum, using the second formula.

We circled in the numerator what we can combine with the formula (the difference of squares) and we circled in the denominator the expressions that are combined with the formula (the difference of cubes).

What do we get as a result? We have x minus 1 in the numerator and denominator. We reduce them, and the uncertainty disappears miraculously.

We write down the answer.

Let's talk a little bit about other uncertainties.

Here is 0 multiplied by infinity.

It is difficult to estimate: it seems that 0 multiplied by anything is 0, on the other hand: infinity ... absorbs any number.

 0 and infinity are competing, and it is not clear what will win, and what will be in the answer.

 We already know how to fight, how to solve the uncertainty of infinity by infinity.

You can see how this uncertainty is easily reduced to the considered uncertainties.

 Then we work with these uncertainties according to the rules.

The uncertainty of infinity minus infinity is also often reduced to the uncertainties discussed above.

 For example, we start calculating the limit, plug 2 for x, and get 1 divided by 0, which is infinity in the theory of limits.

 Minus 3 divided by 0 is also infinity.

 This is the uncertainty of infinity minus infinity.

Everything is quite simple here. What do we do?

We bring these fractions to a common denominator, an additional multiplier for the first fraction is x plus 2, and, the uncertainty immediately disappears. Sometimes there is some uncertainty, for example, infinity divided by 0, or some other; in this case the uncertainty has disappeared.

 The approach is creative when calculating limits, and sometimes you need to think about what to do.

 However, the main uncertainties are 0 divided by 0 and infinity divided by infinity.

Many uncertainties come down to it.