Important limit formula
Evaluating Limit Reference Links:
https://www.sohu.com/a/214347954_507476
Special limit on the e
\[\lim_{x \to \infty} (1+\frac{1}{x})^x = e\]
\[\lim_{x \to 0} (1+ x)^{\frac{1}{x}} = e\]
Power limit on x refers to special functions
\[\lim_{x \to +\infty} x^{\frac{1}{x}} = 1\]
\[\lim_{x \to 0^+} x^x = 1\]
\[\lim_{x \to 0^+} x \ln x =0\]
Special limit with Radical
\[\lim_{n \to \infty} \sqrt[n]{a} = 1\]
\[\lim_{n \to \infty} \sqrt[n]{n} = 1\]
Taylor expansion
1, \ (sin (x) \)
\[sin(x) = x-\frac{1}{6}x^3+O(x)\]
\[ sin(x) = \sum_{i = 0}^{i = n} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \]
2、\(arcsin(x)\)
\[arcsin(x) = x+\frac{1}{6}x^3+O(x)\]
\[ arcsin(x) = \sum_{i = 0}^{i = n} \frac{x^{2n+1}}{(2n+1)!} \]
3 \ (tan (x) \)
\ [Tan (x) = x + \ frac {1} {3} x ^ 3 + O (x + 3) \]
\[ tan(x) = \sum_{i = 0}^{i = n} \frac{x^{2n+1}}{2n+1} \]
4, (have to remember the simplicity derived cumbersome and error-prone) \ (arctan (the X-) \)
\[arctan(x) = x - \frac{1}{3}x^3 + O(x^3)\]
\[ arctan(x) = \sum_{i = 0}^{i = n} (-1)^{2n+1}\frac{x^{2n+1}}{2n+1} \]
5、\(cos(x)\)
\[cos(x) = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + O(x^4)\]
\[ cos(x) = \sum_{i = 0}^{i = n} (-1)^n\frac{x^{2n}}{(2n)!} \]
6, \ (LN (1 + x) \)
\[ln(1+x) = x-\frac{1}{2}x^2 + \frac{1}{3}x^3 + O(x^3)\]
\[ ln(1+x) = \sum_{i = 0}^{i = n} (-1)^n \frac{x^{n+1}}{n+1} \]
7、\(e^x\)
\[e^x = 1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+O(x^3)\]
\[ e^x = \sum_{i = 0}^{i = n} \frac{x^n}{n!} \]
8、\((1+x)^{\alpha}\)
\[(1+x)^{\alpha} = 1+\alpha x +\frac{\alpha(\alpha-1)}{2}x^2 +O(x)\]
\[ (1+x)^{\alpha} = \sum_{i = 0}^{i = n} \frac{C^n_{\alpha}}{n!} \cdot x^n \]
9、\(\frac{1}{1-x}\)
\[\frac{1}{1-x} = 1 + x + x^2 + x^3 + O(x^3)\]
\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \]
10、\(a^x\)
\ [A ^ x = 1 + x \ LN a \]
11、\((1+x)^{\frac{1}{x}}\)
\[(1+x)^{\frac{1}{x}} = e(1-\frac{x}{2} + \frac{11x^2}{24} - \frac{7x^3}{16} \cdots) = e - \frac{x}{2}e + \frac{11x^2}{24}e - \frac{7x^3}{16}e \cdots\]
(Error-prone points) using the equivalent infinitesimal and limit the Taylor expansion of demand conditions
Focus! !
1, do multiplication and division, can
2, add and subtract, only part of the case can verify the feasibility of the method:
For
\ [\ lim a + b \
] with Ru Taile or the equivalent infinitesimal see satisfies \ (\ frac {a} {
b} = \ pm 1 \) is brought can not, or can be brought; e.g. : \
[\ Lim \ X COS -. 1 = (l- \ FRAC X ^ {2} {2}) -. 1 = - \ ^ X FRAC {2} {2} \]
as to bring the \ (\ FRAC {1- \ frac {x ^ 2
} {2}} {- 1} \ neq \ pm1 \) and
\ [\ lim \ sin x - \ tan x \] does not work, because the brought to \ (\ frac { x} {- x} = -1 \)
Classification limit calculation problems
Limit calculation function
1, the difference between the molecular formula for the two
Jane formulated using the squared difference
For example: 1000 question 1.7 \ (\ FRAC {\ sqrt-5X. 1} {- \ sqrt {2x + X ^. 5}} {2-4} \)
2, with the exponential function \ (\ frac {1} { x} \) , or with a \ (\ ln f (x) \) of
(Reciprocal) put down simply because formula
For example:
1.9 1000 question \ (\ lim \ limits_ {x \ to \ infty} e ^ {- x} (1+ \ frac {1} {x}) ^ {x ^ 2} \)
1.11
3, with the integration of
Integration into fractions, eliminating integration with Hospitol
For example:
1000 theme 1.8
4, trigonometric functions can not be calculated with Rutai Le expansion
For example:
1.5 1000 question
5, a median Lagrange's theorem
If \ ([A, B] \) (open interval, can be closed interval) may be turned continuously, then:
$ \ Exists \ epsilon \ in (a, b) $ such that
\[ f'(\epsilon)(b-a) = f(b) - f(a)\] 或 \[f'(\epsilon) = \frac{f(b) - f(a)}{b-a}\]
6, \ (^ \ infty \ 1) calculation of the limit
\ (x \ to \ infty, g (x) ^ {f (x)} \) is equal to \ (e ^ A \)
\[A= f(x)[g(x)-])\]
For example: 1.66
Infinitely smaller than the order
1, when the \ (A \ NEQ 0, K> 0 \) , \ (X \ to 0 \) when \ (F (X) \ SIM AX ^ K \) \ (\ Rightarrow \) \ (X \ to 0 \) when, f (x) is infinitesimal of order k x
2, if the (k> 0 \) \ time, \ (\ Lim \ limits_ {X \ to 0} \ FRAC {F (X)} {X ^ K} \) \ (\ Rightarrow \) \ (X \ to 0, f (x) \) is \ (x \) is \ (k \) infinitesimal
3, if the \ (F (X) = a_0 + a_1x + \ cdots a_k X ^ K \ cdots \) , where \ (a_0 + a_1x + \ cdots A_ {K-. 1} = 0 \) , but \ (a_k \ neq 0 \) , then \ (f (x) \) is infinitesimal of order k x
4, when the \ (x \ to 0 \) , \ (G (x) \) of order n x is infinitesimal, \ (F (x) \) is the m-th order x is infinitesimal, \ [\ int ^ {G ( x)} _ 0 f (t ) dt \] is the x \ ((m + 1) \ cdot n \) infinitesimal
5, if the \ (x \ to 0 \) when \ (F (x) \) and \ (G (x) \) are of order m x infinitesimal and n infinitesimal, and \ (\ lim \ limits_ {x \ to 0} H (x) = a \ NEQ 0 \) , then
. 1) \ (F (x) H (x) \) is the m-th order x infinitesimal
\ (f (x) g ( x) \) is of x \ (m + n \) infinitesimal
2) \ (m> n \) when, \ (F (x) + G (x) \) of order n x is infinitesimal
. 3) \ (m = n \) when, \ (F (x) + G (x) \) is greater than or of order n x n order infinitesimal
Calculation of the limit of the number of columns
1, into a function to count limit
You can then use 'Hospital, Lagrange value theorem
2, the first sum or product
3. Squeeze Rule
1, a simple scaling
n and n-number of no more than n multiplied by the maximum value, the minimum value is not less than n multiplied by
例如：\[\lim\limits_{n \to \infty} (\frac{n}{n^2+1} + \frac{n}{n^2+2} \cdots \frac{n}{n^2+n})\]
A limit is set original;
the number of columns is the maximum value \ (\ {n-FRAC} ^ {n-2}. 1 + \) , the minimum value of \ (\ frac {n} { n ^ 2 + n} \)
\[\frac{n}{n^2+n} \cdot n \leq A \leq \frac{n}{n^2+1} \cdot n\]
\[\Rightarrow \frac{1}{1+\frac{1}{n}} \leq A \leq \frac{1}{1+\frac{1}{n^2}}\]
\[\Rightarrow A=1\]
Adding a limited number of m
例如：\[ \lim\limits_{n \to \infty} \sqrt[n]{a_1^n + a_2^n +\cdots a_m^n}, 0 \leq a_i(i = 1,2,3, \cdots m)\]
This problem is to be noted, to find the maximum value , the maximum value is greater than, less than the maximum value and the m
Provided the original and the number of columns is A, a maximum value of \ (a_1 = max (a_1, a_2, \ cdots, a_m) \) then
\ [a_1 ^ n \ leq A \ leq a_1 ^ n \ cdot m \]
\[\Rightarrow a_1 \leq \sqrt[n]{A} \leq a_1 \cdot m^{\frac{1}{n}}\]
\[\Rightarrow \lim\limits_{n \to \infty} A = a_1=max(a_1, \cdots ,a_m)\]
Important conclusion:
the form \ [\ lim \ limits_ {n \ to \ infty} \ n sqrt [] {a_1 ^ n + a_2 ^ n + \ cdots a_m ^ n} = max (a_1, a_2, \ cdots, a_m) \ ] m limited
例1：\[\lim\limits_{n \to \infty} \sqrt[n]{2020 + 2^n+3^n+4^n}=4\]
Example 2: \ [\ Lim \ limits_ {n-\ to \ infty} \ n-sqrt [] {. 1 + X ^ n-+ (\ FRAC {X ^ 2} {2}) ^ n-} \] (remember classification discussed)
4, monotone bounded guidelines
5, the number of columns of the structuring method
Seeking application of the limit
1, intermittent points
the first sort
It may be short discontinuous point
\ (x = x_0 \) no definition at
Jump discontinuity point
limit ranging from about
The second category
Infinite discontinuity points
without limits and no boundaries
Oscillation break point
without limits there are boundaries
2, function curve asymptotes Seeking
Vertical asymptotes
- Find someone undefined point \ (x_0 \)
- The demand limit point \ [\ lim_ {x \ to \ infty} f (x) = \ \ infty] limit exists, \ (X = x_0 \) is the vertical asymptote
Horizontal asymptote
Of Limit \ [\ lim_ {X \ to \ PM \ infty} F (X) = A \] \ (A \) whether there
Exists \ (y = A \) is the horizontal asymptote
Oblique asymptote
If \ (y = f (x) \) is satisfied:
- \ [\ lim_ {X \ to \ infty} \ FRAC {F (X)} = {X} K \] \ (K \) present
- \ [\ lim_ {X \ to \ infty} (F (X) -kx) = B \] \ (B \) present
There swash asymptote \ (y = kx + b \ )