The value of any expression aspires to any limit which size is a constant. Tasks on limits very often meet it is aware of the mathematical analysis. Their decision demands existence of a number of specific knowledge and skills.
Instruction
1. A limit is called some number to which the variable variable or value of expression aspires. Usually variable or functions tend either to zero, or to infinity. At the limit equal to zero, size is considered infinitesimal. In other words, sizes which changes also approach zero are called infinitesimal. If the limit strives for infinity, then it is called an infinite limit. Usually he registers in a look: lim x=+ ∞.
2. Limits have a number of properties some of which represent axioms. The main of them are given below. - one size has only one limit; - the limit of a constant is equal to the size of this constant; - 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 * a constant multiplier can be taken out by lim y-for the sign of a limit: lim(Cx) = C * lim x, where C=const; - the limit of private is equal private limits: lim (x/y) =lim x/lim y.
3. In tasks both numerical expressions, and derivatives of these expressions meet limits. It can look, in particular, as follows: lim xn=a (at n →∞). The example of a simple limit is given below: lim 3n +1/n+1n →∞. For the solution of this limit divide all expression into n of units. It is known that if unit is divided into some size n →∞, then the limit 1/n is equal to zero. Fairly and the return: if n→0, then 1/0= ∞. Having divided all example into n, write down it in the look given below and receive the answer: lim 3+1/n/1+1/n=3n →∞.
4. At the solution of tasks on limits there can be results which are called uncertainty. In such cases apply the rules Lopitalya. For this purpose make repeated differentiation of function which will give an example in such form in which it could be solved. There are two types of uncertainty: 0/0 and ∞/∞. The example of c uncertainty can look, in particular, the following obrashchy: lim 1-cosx/4x^2= (0/0) =lim sinx/8x=(0/0) =lim cosx/8=1/8x→0.
5. The second type of uncertainty uncertainty of a look ∞/∞ is considered. It often meets, for example, at the solution of logarithms. Below the example of a limit of a logarithm is shown: lim lnx/sinx= (∞/∞) =lim1/x/cosx=0x → ∞.