How to judge when the Taylor formula is found to the limit to the first term

The statement of the netizen upstairs is not appropriate:

1. There is no numerator denominator.

The highest power is the same.

According to this statement, if the numerator and denominator are odd functions.

One is the even function.

will fall into the point where there is no answer.

2. There are no more items to try to be safe.

Too much is unnecessary, a waste of time, and more importantly, it does not generate intuition and misleads judgment.

In fact, as long as it arrives, it must also reach the first item that cannot be offset.

No need to add too much to it, no need to ask for trouble, no need to waste time.

If you have any questions, please feel free to ask them.

Respectfully, those who have the right to elect and certify "Professional Answers" do not choose to certify as "Professional Answers".

Even if mine is full of mistakes, I can't get pertinent criticism from netizens, which is very unfair.

Please be considerate and do not elect for certification. Thank you!

In general, it is necessary to observe the numerator and denominator of the function to find the limit, and if only the numerator is required, it should not be lower than the highest power of the denominator. Vice versa.

If both the numerator and the denominator are required, in this case, some terms may be canceled out by other functions of addition and subtraction, until the numerator and denominator are comparable.

Generally, it can be ignored when calculatingHigher-order infinitesimalsThat's it. Let's sayDenominatorThere is an x 2, and your numerator is followed by o (x 2) after x 2, so that the higher order infinitesimal tends to zero when the calculation is recalculated, and it does not affect the calculation result. This level is fine.

Taylor's formula. is a formula that describes the value of a function near a point with information about it. If the function satisfies certain conditions, Taylor's formula can approximate the function by using the coefficients of each derivative of the function at a certain point to construct a polynomial.

Taylor's formula takes its name from the English mathematician Brooke Taylor, who first described it in a letter in 1712. Taylor's formula is a chain of approximations that are often used to study the properties of complex functions, and is also an important application of functional differentiation.

Historical development. Taylor's formula is advanced mathematics.

It approximates some complex functions as simple polynomial functions.

The simplification of the Taylor formula makes it a powerful tool for analyzing and researching many mathematical problems.

Brook Taylor, one of the most prominent mathematicians of the English Newtonian school in the early 18th century, wrote in 1715 in his book The Incremental Method of Positive and Negative, in which he stated his famous theorem in a letter to his teacher Machin in July 1712, which was Taylor's theorem.

I think there is a 2:1 where x tends to 0; 2. It is more convenient to use Taylor's formula when it is complicated to use Lopida's rule to derive it, because Taylor's formula is a polynomial operation and an infinitesimal operation.

Different types of functions are generally solved by Taylor's formula when adding and subtracting, such as x-sinx, x-arctanx, etc., which are converted into polynomials and solved by Robida infinitesimals.

Ratios can be used, mainly because it is inconvenient to see the square, especially when mixing complex types in polynomial ratios.

Taylor's formula to find the limit, with a need to look at the question setting, some questions can be answered with 3 items, while some questions require n items.

If the function f(x) has a derivative of the nth order on a closed interval [a,b] containing x0 and a derivative of (n+1) on the open interval (a,b), then for any point x on the closed interval [a,b], the following equation is true:

where the nth derivative of f(x) is described, and the polynomial after the equal sign is called Taylor's formula for the function f(x) at x0, and the remaining rn(x) is the remainder of Taylor's formula, which is the higher-order infinitesimal of (x-x0)n.

Can only be usedTaylor's formula. In the case of finding the limit: there is generally a form of the sum or difference of more than two functions, and the self-variable x is 0 when finding the limit, so that the higher power of x can be replaced by 0 when x 0.

The case where the Taylor formula is used to find the limit is that the limit expression found by the over-the-top spring worm contains trigonometric functions.

Power function, exponential function.

When the equations of logarithmic functions are added and subtracted, or the composite functions of these functions are used as the numerator or denominator, it is not easy to find the limit by other methods, so we should think of using the Taylor formula to find the limit.

This is in the pair function

One of the formulas commonly used for local linearization. Geometrically, it is a tangent approximation instead of a curve. However, such an approximation is rather crude and only has approximate significance in the vicinity of the point.

In order to improve the above shortcomings and make the approximate substitution more precise, mathematicians have developed the Cauchy median theorem.

On the basis of Taylor, Taylor's median value theorem (Taylor Gongsen limb pose) is derived.

Let y=x sinx.........Empty .........1）

Take the logarithm of both sides to get the following

lny=sinx*lnx

The derivation of x on both sides yields: (1 y)*y = sinx x+lnx*cosx(2).

From (1) and (2), we get y = (sinx x + lnx * cosx) * x (sinx).

In the process of a certain variable in a function, in the process of the eternal change of this variable becoming larger (or smaller), the closing loss and leakage gradually approach a certain definite value a and "can never coincide to a" ("can never be equal to a, but taking equal to a' is enough to obtain high-precision calculation results"), the change of this variable is artificially defined as "always approaching without stopping", and it has a "tendency to constantly get extremely close to point a".

There are basic ways to find the limit.

1. In the fraction, the numerator and denominator are divided by the highest order of the sedan rotten, and the infinitesimal is calculated as infinitesimal, and the infinitesimal is directly substituted with 0;

2. When the infinity root formula subtracts the infinite root formula, the molecule is rationalized;

3. Apply Lopida's law, but the application condition of Lopida's law is to become infinitely larger than infinite, or infinitesimal than infinitesimal, and the numerator denominator must also be a continuous derivative function.

4. Use McLaurin (McLaullin) series, and it is generally mistranslated as Taylor (Taylor) in China.