In textbooks according to the mathematical analysis considerable attention is paid to methods of calculation of limits of functions and the sequences. There are ready rules and methods, applying which, it is possible to solve with ease even rather complex problems on limits.
Instruction
1. In the mathematical analysis there are concepts of limits of the sequences and functions. When it is required to find a sequence limit, it is written down as follows: lim xn=a. In such sequence of the sequence of xn n to infinity aspires to a, and. The sequence is usually presented in the row form, for example: x1, x2, x3..., xm..., xn.... The sequences are subdivided into increasing and decreasing. For example: xn=n^2 - increasing posledovatelnostyn=1/n - decreasing posledovatelnosttak, for example, a limit of the sequence of xn=1/n^2 is equal: lim 1/n^2=0x →∞ This limit is equal to zero as n →∞, and the sequence 1/n^2 tends to zero.
2. Usually the variable x aspires to a final limit of a, and, x constantly approaches a, and size a is constant. It is written down as follows: limx =a, at the same time, of n can also aspire both to zero, and to infinity. There are infinite functions, for them the limit strives for infinity. In other cases when, for example, function describes delay of the course of the train, it is possible to speak about the limit tending to zero. Limits have a number of properties. As a rule, any function has only one limit. This main property of a limit. Other their properties are listed below: * The limit of the sum is equal to the sum of limits: lim(x+y)=lim x+lim y * the Limit of the work is equal to the work of limits: lim(xy)=lim x*lim y * the Limit of private is equal private from limits: lim (x/y) =lim x/lim y * take out the Constant multiplier for the sign of a limit: lim(Cx) = C lim hesli is given function 1/x in which x →∞, its limit is equal to zero. If x→0, a limit of such function is equal ∞. For trigonometrical functions there are exceptions of these rules. As the sin x function always aspires to unit when it approaches zero, for it the identity is fair: lim sin x/x=1x→0
3. In a number of tasks, functions at which calculation of limits there is an uncertainty - a situation at which the limit cannot be calculated meet. Application of the rule Lopitalya becomes the only exit from such situation. There are two types of uncertainty: * uncertainty of a look 0/0 * uncertainty of a look / k to an example, the limit of the following look is given: lim f (x)/l (x), and, f(x0)=l(x0)=0. In that case, there is an uncertainty of a look 0/0. For the solution of such task both functions subject to differentiation then find a result limit. For uncertainty of a look 0/0 limit are equal: lim f (x)/l (x) '(x)/l' (x) (at x→0) Same ruled =lim f fairly and for uncertainty like ∞/∞. But in this case fairly following equality: f (x) =l (x)= ∞ by means of the rule Lopitalya can find values of any limits in which uncertainty appear. An indispensable condition besides - lack of mistakes when finding derivatives. So, for example, the derivative of function (x^2)' is equal 2x. From here it is possible to draw a conclusion that: f' (x) =nx^ (n-1)