How to solve the limits?
In the course of mathematical analysis, a sufficiently large period of time is allocated for studying the 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, when properly applied, can help to solve rather difficult tasks with limits.
The concepts of how to solve the limits of functions, as well as the limits of sequences, were introduced into the mathematical analysis. If it is necessary to calculate the limit of the sequence, then the record of this example looks like this: lim xn = a. From this sequence, it is clear that xn tends to a. In turn, n, on the contrary, tends to infinity. Most often, sequences are represented as rows, such as, for example, p1, p2, p3 ..., pm, ..., pn .... All sequences are usually divided into two groups: decreasing sequences, and also increasing sequences.
How to solve limits: formulas
Most often, a quantity that is variable, for example, x tends to a finite limit, which is the quantity a.The value of x is constantly approaching the value of a, at which time as the value of a remains constant. Writing this complex definition is very simple: limx = a. In this case, n may tend to infinity, and to zero. There are special functions called infinite. In them, the limit also tends to infinity. If another function is considered that describes the slowing down of something, then there is a sense to talk about the limit, which will tend to zero.
All chapels have their own specific number 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 limits are related to their definition and problem solving. Also, students should pay attention to the topic of how to solve limits with roots.
- The limit of the sum is equal to the sum of all limits: lim (x + y) = lim x + lim y.
- The limit of the quotient is equal to the quotient of all limits: lim (x / y) = lim x / lim y.
- The limit of the product is equal to the product of all limits: lim (xy) = lim x * lim y.
- A constant factor can be taken out of the limit sign without losing the integrity of the problem: lim (Cx) = C lim x.
If the task is given the function 1 / x, for which x tends to infinity, then the limit of this function will be equal to zero.If x tends to zero, then, accordingly, the limit of this function will be equal to infinity.
There are special exceptions to these rules for trigonometric functions. The sine function x always tends to one. When this function approaches zero, the given identity becomes valid for it: lim sin x / x = 1.
There are also problems of this kind, where in the process of calculating limits a certain situation appears, called uncertainty. Uncertainty is a situation in which the calculation of the limit is impossible. The only way to solve such problems is to use the L'Hôpital rules. All uncertainties can be divided into two categories: the uncertainty of the form zero is divided by zero, and the uncertainty of the form infinity divided by infinity. If we consider the limit lim f (x) / l (x), when f (x0) = l (x0) = 0, then in this case there is a clear occurrence of the uncertainty of the form zero divided by zero. In order to properly solve this problem, both functions must first be differentiated and then the result limit calculated. If we consider the uncertainty of the form of zero divided by zero, then the limit of the function will be equal to: lim f (x) / l (x) = lim f '(x) / l' (x) (for x tending to zero).
There are special services where you can solve the online limit of any complexity. For example, the site "" offers a standard calculation of the limit of functions using a special calculator. If it is necessary to calculate the limit of the sequence, it is better to contact the online calculator of limits on the site "".