Main uncertainties of limits and their disclosure. L'Hopital's rule: theory and examples of solutions Any number to an infinite power

Type and species uncertainty are the most common uncertainties that need to be disclosed when solving limits.

Most of the limit problems encountered by students contain just such uncertainties. To reveal them or, more precisely, to avoid uncertainties, there are several artificial techniques for transforming the type of expression under the limit sign. These techniques are as follows: term-wise division of the numerator and denominator by the highest power of the variable, multiplication by the conjugate expression and factorization for subsequent reduction using solutions to quadratic equations and abbreviated multiplication formulas.

Species uncertainty

Example 1.

n is equal to 2. Therefore, we divide the numerator and denominator term by term by:

.

Comment on the right side of the expression. Arrows and numbers indicate what fractions tend to after substitution n meaning infinity. Here, as in example 2, the degree n There is more in the denominator than in the numerator, as a result of which the entire fraction tends to be infinitesimal or “super-small.”

We get the answer: the limit of this function with a variable tending to infinity is equal to .

Example 2. .

Solution. Here the highest power of the variable x is equal to 1. Therefore, we divide the numerator and denominator term by term by x:

.

Commentary on the progress of the decision. In the numerator we drive “x” under the root of the third degree, and so that its original degree (1) remains unchanged, we assign it the same degree as the root, that is, 3. There are no arrows or additional numbers in this entry, so try it mentally, but by analogy with the previous example, determine what the expressions in the numerator and denominator tend to after substituting infinity instead of “x”.

We received the answer: the limit of this function with a variable tending to infinity is equal to zero.

Species uncertainty

Example 3. Uncover uncertainty and find the limit.

Solution. The numerator is the difference of cubes. Let’s factorize it using the abbreviated multiplication formula from the school mathematics course:

The denominator contains a quadratic trinomial, which we will factorize by solving a quadratic equation (once again a link to solving quadratic equations):

Let's write down the expression obtained as a result of the transformations and find the limit of the function:

Example 4. Unlock uncertainty and find the limit

Solution. The quotient limit theorem is not applicable here, since

Therefore, we transform the fraction identically: multiplying the numerator and denominator by the binomial conjugate to the denominator, and reduce by x+1. According to the corollary of Theorem 1, we obtain an expression, solving which we find the desired limit:

Example 5. Unlock uncertainty and find the limit

Solution. Direct value substitution x= 0 into a given function leads to uncertainty of the form 0/0. To reveal it, we perform identical transformations and ultimately obtain the desired limit:

Example 6. Calculate

Solution: Let's use the theorems on limits

Answer: 11

Example 7. Calculate

Solution: in this example the limits of the numerator and denominator at are equal to 0:

; . We have received, therefore, the theorem on the limit of the quotient cannot be applied.

Let us factorize the numerator and denominator in order to reduce the fraction by a common factor tending to zero, and, therefore, make it possible to apply Theorem 3.

We expand the square trinomial in the numerator using the formula, where x 1 and x 2 are the roots of the trinomial. Having factorized and denominator, we reduce the fraction by (x-2), then apply Theorem 3.

Answer:

Example 8. Calculate

Solution: When the numerator and denominator tend to infinity, therefore, when directly applying Theorem 3, we obtain the expression , which represents uncertainty. To get rid of uncertainty of this type, you should divide the numerator and denominator by the highest power of the argument. In this example, you need to divide by X:

Answer:

Example 9. Calculate

Solution: x 3:

Answer: 2

Example 10. Calculate

Solution: When the numerator and denominator tend to infinity. Let's divide the numerator and denominator by the highest power of the argument, i.e. x 5:

=

The numerator of the fraction tends to 1, the denominator tends to 0, so the fraction tends to infinity.

Answer:

Example 11. Calculate

Solution: When the numerator and denominator tend to infinity. Let's divide the numerator and denominator by the highest power of the argument, i.e. x 7:

Answer: 0

Derivative.

Derivative of the function y = f(x) with respect to the argument x is called the limit of the ratio of its increment y to the increment x of the argument x, when the increment of the argument tends to zero: . If this limit is finite, then the function y = f(x) is said to be differentiable at x. If this limit exists, then they say that the function y = f(x) has an infinite derivative at point x.

Derivatives of basic elementary functions:

1. (const)=0 9.

3. 11.

4. 12.

5. 13.

6. 14.

Rules of differentiation:

a)

V)

Example 1. Find the derivative of a function

Solution: If the derivative of the second term is found using the rule of differentiation of fractions, then the first term is a complex function, the derivative of which is found by the formula:

, Where , Then

When solving the following formulas were used: 1,2,10,a,c,d.

Answer:

Example 21. Find the derivative of a function

Solution: both terms are complex functions, where for the first , , and for the second , , then

Answer:

Derivative applications.

1. Speed ​​and acceleration

Let the function s(t) describe position object in some coordinate system at time t. Then the first derivative of the function s(t) is instantaneous speed object:
v=s′=f′(t)
The second derivative of the function s(t) represents the instantaneous acceleration object:
w=v′=s′′=f′′(t)

2. Tangent equation
y−y0=f′(x0)(x−x0),
where (x0,y0) are the coordinates of the tangent point, f′(x0) is the value of the derivative of the function f(x) at the tangent point.

3. Normal equation
y−y0=−1f′(x0)(x−x0),

where (x0,y0) are the coordinates of the point at which the normal is drawn, f′(x0) is the value of the derivative of the function f(x) at this point.

4. Increasing and decreasing functions
If f′(x0)>0, then the function increases at the point x0. In the figure below the function is increasing as x x2.
If f′(x0)<0, то функция убывает в точке x0 (интервал x1If f′(x0)=0 or the derivative does not exist, then this criterion does not allow us to determine the nature of the monotonicity of the function at the point x0.

5. Local extrema of a function
The function f(x) has local maximum at the point x1, if there is a neighborhood of the point x1 such that for all x from this neighborhood the inequality f(x1)≥f(x) holds.
Similarly, the function f(x) has local minimum at the point x2, if there is a neighborhood of the point x2 such that for all x from this neighborhood the inequality f(x2)≤f(x) holds.

6. Critical points
Point x0 is critical point function f(x), if the derivative f′(x0) in it is equal to zero or does not exist.

7. The first sufficient sign of the existence of an extremum
If the function f(x) increases (f′(x)>0) for all x in some interval (a,x1] and decreases (f′(x)<0) для всех x в интервале и возрастает (f′(x)>0) for all x from the interval )