How to solve the limits?

In the course of mathematical analysis,a large amount of time is allocated to studying techniques of how to solve limits, both for functions and for sequences. At the moment, there are a number of ready-made methods and rules that, if properly applied, can help solve fairly difficult tasks with limits.

In the mathematical analysis were introduced conceptsHow to solve the limits of functions, as well as the limits of sequences. If it is necessary to calculate the limit of a sequence, then the record of this example looks like this: lim xn = a. It can be seen from this sequence that xn tends to a. In turn, n tends to infinity. Most often, sequences are represented as series, such as, for example, p1, p2, p3 ..., pm, ..., pn .... All sequences are divided into two groups: decreasing sequences, and increasing sequences.

How to solve limits: formulas

Most often a quantity that is variable,for example, x tend to a finite limit, which is the value of a. In this case, the quantity x constantly approaches the value a, in which time as a remains constant. The record of this complex definition is very simple: limx = a. In this case, n can tend to infinity, and to zero. There are special functions that are called infinite. In them the limit also tends to infinity. If, however, another function is considered that describes a slowdown of something, then it makes sense to talk about a limit that will tend to zero.

All the chapels have their own specific set of properties. Most often, one function can have only one limit. This is the most important and most important property of the limits. All other properties of the limits are related to their definition and solution of problems. Also, students should pay attention to the topic of how to solve the limits with roots.

1. The limit of the sum is equal to the sum of all limits: lim (x + y) = lim x + lim y.
2. The limit of the quotient is equal to the quotient of all limits: lim (x / y) = lim x / lim y.
3. The limit of the product is equal to the product of all limits: lim (xy) = lim x * lim y.
4. A constant factor can be taken as a limit sign without loss of the integrity of the problem: lim (Cx) = C lim x.

If the job is provided with 1 / x, forwhich tends to infinity, then the limit of this function will be zero. If, however, x tends to zero, then, respectively, and the limit of this function will be equal to infinity.

There are special exceptions to these rulesfor trigonometric functions. The sine function x always tends to unity. When this function approaches zero, the following identity becomes valid for it: lim sin x / x = 1.

There are also such problems, where inThe process of calculating the limits introduces a certain situation, called uncertainty. Uncertainty is the situation in which the calculation of the limit is impossible. The only way to solve such problems is to use L'Hospital's rules. All uncertainties are divided into two categories: the uncertainty of the form zero divide by zero, and also the uncertainty of the form infinity divide by infinity. If we consider the limit lim f (x) / l (x), when f (x0) = 1 (x0) = 0, then in this case an explicit occurrence of uncertainty of the form is zeroed. In order to correctly solve this problem, both functions must first be differentiated, and then the limit of the result is calculated. If we consider zero-degree uncertainty, then the limit of the function is equal to lim f (x) / l (x) = lim f (x) / l (x) (for x tending to zero).

There are special services where you can solvethe limit of online of any complexity. For example, the "Calculate Online Limit" website offers a standard function limit counting using a special calculator. If it is necessary to calculate the limit of the sequence, it is better to refer to the online calculator limits on the site "Solving the limits online."