When x is approaching 0, find the limit, and when x is approaching 0, does the limit exist?

It's difficult, I can only do a few steps, you see!

First of all, the original formula is an infinitive of type 0 0 and satisfies the conditions of Lopida's law, so the original formula = lim 1 (using the Lopida rule once).

lim/1lim[ln(1+x)e^(1/x)*ln(1+x)]/x+lim[ln(1+x)/x]*lim[(e^x)/(x+1)]

lim[ln(1+x) x]*lim[e (1 x)*ln(1+x)]+lim[ln(1+x) x]*lim[(e x) (x+1)] makes use of Lopida's rule once).

lim[e^(1/x)*ln(1+x)]

The rest is not known! It seems that it can be re-converted into fractions and continue to use the Lopida law.

The molecule can be reduced to exp[(1 x)ln(1+x)]-e=e* The numerator is infinitesimal, and with the equivalent infinitesimal, the numerator = e*[(1 x)ln(1+x)-1].

Original formula = lim e*[ln(1+x)-x] (x 2) Lobida rule = lim e*[1 (1+x)-1] (2x)=lim e*x [2x(1+x)]=e 2

If x is tending to 0, the limit does not exist. When x 0, the limit is 1. When x 0, the limit is 1.

The so-called limit idea refers to "a mathematical idea that uses the concept of limit to analyze and solve problems". For the unknown quantity to be examined, first try to correctly conceive another variable related to its change, and confirm that the 'influence' trend of this variable through the infinite change process is that the result of the very precise leakage is approximately equal to the unknown quantity sought; Using the limit principle, the results of the unknown quantities under investigation can be calculated.

The limit idea is calculus.

The basic idea is mathematical analysis.

A number of important concepts such as the continuity of functions, derivatives (to 0 gives a maximum), and definite integrals are defined with the help of limits. Min Chi Yu.

The above information refers to the encyclopedia - Limit.

x→0，1-cosx~x^2/2

Commonly used infinitesimal substitution formula:

When x 0.

sinx~x

tanx~x

arcsinx~x

arctanx~x

1-cosx~1/2x^2

a^x-1~xlna

e^x-1~x

ln(1+x)~x

1+bx)^a-1~abx

1+x)^1/n]-1~1/nx

loga(1+x)~x/lna

Limit. Basic concepts of mathematical analysis. It refers to the trend of change and the value of the trend of the variable (limit value) in the process of a certain change, from the general gradually stable in general.

The limit method is the basic method used by mathematical analysis to study functions, and the various basic concepts of analysis (continuous, differential, integral, and series) are based on the concept of limits, and then all the theories, calculations, and applications of analysis are presented. Therefore, a precise definition of the concept of limit is necessary, and it is the fundamental question of whether the theory and calculation involved in the analysis are reliable.

Historically, it was Cauchy, A-l.First, a general definition of the limit is given more clearly.

He said, "When all the series of values for the same variable are infinitely close to a fixed value, and the final difference from it is as small as it is" (Tutorial of Analysis, 1821), this fixed value is called the limit of the variable.

Subsequently, Weierstrass (Weierstrass, K. Waierstrass).(According to the idea of Haru Tezhao, the definition of the limit of strict quantification is the -δ definition or - definition used in mathematical analysis.) Since then, there have been practical criteria for judging various limit problems.

The concept of limits is equally important in other disciplines of analytics, and there are some generalizations in disciplines such as functional analysis and point-set topology.

1. When x is infinite, the specific answer is as follows.

2. Rules. Whenever you want to find the limit of the round stupid hail, the trend and infinity, come up to look at the numerator code denominator, only look at the higher power, the highest power in the numerator is infinity (does not exist), the highest power of the orange sail in the denominator is 0, if the numerator denominator is the same, it is equal to the coefficient in front of them. x tends to 0 to see the lowest power.

x approaches Yu's ridge 0 and finds the equivalent infinitesimal quantity of x+sinx.

x+sinx~x+x=2x

That is, the vertical field is scattered. x+sinx~2x

1.As long as a specific value can be calculated after substitution, it can be substituted;

2.If you substitute it, you can't get a specific value, but you can get itInfinityThe limit is infinity, whether it is negative or negative, the limit does not exist. The limit does not exist, and it is also a formula.

3.If you substitute for the aftermath of do, what you get isInfinitiveThere are seven kinds of infinitives, which cannot be substituted, but must be calculated by a special method of limit calculation, and cannot be simply substituted directly.

"Limit" is a branch of mathematics – calculus.

The basic concept of "limit" in the broad sense means "infinitely close and never reachable". The "limit" in mathematics refers to the process in which a variable in a function is gradually approaching a certain definite value a and "can never coincide to a" ("can never be equal to a, but taking and equaling a' is enough to obtain high-precision calculation results) in the process of a variable in a function, which is artificially defined as "always approaching without stopping", and it has a "tendency to constantly get extremely close to point a".

Limit is a description of a "state of change". The value a that this variable is always approaching is called the "limit value" (which can also be represented by other symbols).

The above is a popular description of the connotation of "limit", and the strict concept of "limit" was eventually developed by Cauchy and Weierstras.

and others strictly elaborate.

When x tends to 0, the number orange, the denominator x tends to 0, and the exponent 1 x in the numerator tends to infinity, so this is an infinitive of "0 infinity". It can be solved using Lopida's rule:

Reduce the function to f(x) =e (1 x) xf(x) =e (1 x) x

f'(x) =e^(1/x) /x^2) -e^(1/x) /x^2)

Let x tend to 0, and get f'(x) =1, so the limit of the original formula is:

lim(x->0) e (1 x) x = lim(x->0) [1 f(x)] lim(x-> file pei 0) (x line biwei e (1 x)) 0

So (e (1 x)) x has a limit of 0 when x tends to 0.

The reason is as follows, because when x---0 is sailing, the denominator is not 0, that is to say, 0 is in the definition domain of the clan, so it can be directly brought in, that is, 1 1 = 1 so the limit is 1

Use the formula: a n-1=(a-1)*[1+a+a+a+a+.]a^(n-1)]

The numerator and denominator are multiplied and eliminated at the same time to [1+a+a+a+.].a (n-1)], where a = (1+x) (1 n).

The basic idea or way to find the limit is to find a way to reduce the variable that causes the denominator to be 0, generally using a large number of cavity kernel factorization, and to remember the commonly used limit formula. In this problem, the formula is used to reduce the x that causes the denominator to be 0.