Summary of advanced mathematics question types

1. Functions, Limits and Continuity

(1) Determination of function boundedness, monotonicity, periodicity and odd-evenness

1. Monotonicity

2. Parity

Common odd functions: $sinx，tanx，arcsinx，arctanx，ln\frac{1-x}{1+x}，ln(x+\sqrt{1+x^2})，\frac{e^x-1}{e^x+1}，f(x)-f(-x)$

Common even functions: $x^2，|x|,cosx，f(x)+f(-x)$

3. Periodicity

! $Iff\left(x\right)hasperiodT,thenf(ax\_+b)with\frac{T}{|a|}forthe\; period$

4. Boundedness

$|sinx|\le 1，|cosx|\le 1，|arcsinx|\le \frac{}{2}，|arccosx|\le Pi$

Supplement: Common functions
1）$y=\arctan x$

2） $y=\arcsin x$

3）$y=\arccos x$

4) Sign function:
y = sgnx=$$\{\begin{array}{l}-1，x<0\\ 0，x=0\\ 1，x>0\end{array}$$
5) Rounding function: $y=[x]$
$x-1<\left[x\right]\le x$

(2) Composite function

Compounding is possible only when the intersection of the domain of the outer function and the value domain of the inner function is not empty.

(3) The concept and nature of limit

1. The concept of limits

1) Sequence limit

$\forall ϵ>0，\exists N>0,whenn>WhenN,there\; isalways|x{}_n-a|<ϵ,then\; it\; is\; called\underset{x\to \infty }{\lim }{x}_n=a$

！$b<a,\exists N,thisn>WhenN,x_n>b$
！ $c>a,\exists N,thisn>WhenN,x_n<c$
! $The\; limitofthe\; sequencelimit\left\{x\right\}hasnothing\; to\; dowith\; thepreviousfiniteterm$
! $Overalllimit\exists \Rightarrow Partiallimit\exists$
!$\_$$Allpartiallimits\exists \Rightarrow Overalllimit\exists$

2) Function limit

(1) When the independent variable approaches infinity:

${\lim }_{x\to +\infty }$

$\forall ϵ>0，\exists X>0,whenx>WhenX,thereisalways|f\left(x\right)-A|<ϵ,then\; it\; is\; called\underset{x\to +\infty }{\lim }f(x)=a$

${\lim }_{x\to -\infty }$

$\forall ϵ>0，\exists X>0,whenx<When-X,there\; isalways|f(x)-A|<ϵ,then\; it\; is\; called\underset{x\to -\infty }{\lim }f(x)=a$

${\lim }_{x\to \infty }$

$\forall ϵ>0，\exists X>0,when|x|>WhenX,thereisalways|f\left(x\right)-A|<ϵ,then\; it\; is\; called\underset{x\to \infty }{\lim }f(x)=a$

(2) The independent variable tends to a finite value:

$\forall ϵ>0，\exists \delta >0,when0<|x-x_0|>When\delta ,there\; isalways|f(x)-A|<ϵ,then\; it\; is\; called\underset{x\to \infty }{\lim }f(x)=A$

! $x\to x_o,Butx\ne x_o【i.e.\underset{x\to \infty }{\lim }f(x)\; hasnothing\; to\; dowithwhetherf(x\left)exists\right]$

2. Nature of limits

1) General properties

(1) Boundedness

Convergence $\to$ There is a limit, but not vice versa

(2) Number protection

(3) Relationship with infinitesimal

$\lim f(x)=A\leftrightarrow f\left(x\right)=A+\alpha \left(x\right),\left[where\lim a\right(x)=0】$

2) Existence criteria

(1) Clamping criterion

(2) Monotone bounded criterion

3. Extreme algorithm

(1) Rational operation rules

若 $limf(x)=A,limg\left(x\right)=B$ , then

• $lim(f(x)\pm g(x))=limf\left(x\right)\pm g(x)$
• $lim(f(x)\times g(x))=f(x)\times g(x)$
• $lim\frac{f(x)}{g\left(x\right)}=\frac{limf(x)}{limg\left(x\right)}，(B\ne 0)$

(2) Attention

• exists $\pm$ does not exist$=$ does not exist
• Does not exist $\pm$ does not exist$=$ not necessarily
• exists $\times ÷$ does not exist$=$ not necessarily
• Does not exist $\times ÷$ does not exist$=$ not necessarily

(3) Common conclusions

• $limf(x)=A\ne 0\to limf(x)g(x)=Alimg\left(x\right)$
  means:the limit of the non-zero factor can be found first

• $lim\frac{f(x)}{g(x)}$Existence, $limg\left(x\right)=0\to limf(x)=0$

• $lim\frac{f\left(x\right)}{g\left(x\right)}=A$，$limf\left(x\right)=0\to limg\left(x\right)=0$

4. Mugless Small

1) Infinitesimal order: higher order, k order, same order, equivalent
2) Infinitesimal properties

• The sum/product of a finite number of infinitesimals is still infinitesimal
• The product of an infinitesimal and a bounded quantity is still infinitesimal

5. Mugendai

1) Common infinity comparisons:

To x $\to +\infty$时， $l{n}^{\alpha }x\ll x^b\ll a^x$

Ton $\to \infty$时，$l{n}^{\alpha }n\ll n^b\ll a^n\ll n!\ll n^n$

2) Properties of infinity:

• The product of two infinities is still infinity
• The sum of infinity and bounded variables is still an infinite amount

3) The relationship between infinity and unbounded variables:

• infinite amount $\to$ Unbounded variable, but not vice versa [$(-1{)}^n$】

4) The relationship between infinity and infinitesimal:

• $f\left(x\right)$ is infinitesimal, and$f(x)\ne 0$ , then$\frac{1}{f\left(x\right)}$is $\infty$
  ! If$f\left(x\right)\ne 0$ , then$\frac{1}{f\left(x\right)}$meaningless

(4) Left and right limits

1. The limit of the piecewise function at the dividing point
2. $e^{\infty }$ type limit
3. $arctan\infty$ type limit

(5) Find the limit

1. Find the limit of the sum\product of n terms

1) Find the sum formula first, then find the limit
2) Pinch theorem: The sum of n terms fails, and the degree of the numerator and denominator is not equal
3) Definite integral definition: The degree of the numerator and denominator are both equal, and the denominator is one more than the degree of the numerator.

2. Finding limits indefinitely

（1） $\frac{0}{0}$

$\frac{0}{0}$: $\{\begin{array}{l}Equivalenttoinfinitesimal\\ Lópida\text{'}sLaw:\_\frac{0}{0}or\frac{\infty }{\infty }\\ Taylor\end{array}$

Supplement: Equivalent to infinitesimal

• $sinx，tanx，arcsinx，arctanx，ln(1+x)，e^x-1\sim x$
• $\frac{a^x-1}{x}\sim lna$
• $(1+x{)}^a-1\sim \alpha x$
• $(1+a(x{)}^{b\left(x\right)})-1\sim a(x\left)b\right(x)（a(x)\to 0，\alpha (x\left)\beta \right(x)\to 0）$
• $1-cosx\sim \frac{1}{2}{x}^2$
• $x-ln(1+x)\sim \frac{1}{2}{x}^2$
• $x-sinx\sim \frac{1}{6}{x}^3$
• $arcsinx-x\sim \frac{1}{6}{x}^3$
• $tanx-x\sim \frac{1}{3}{x}^3$
• $x-arctanx\sim \frac{1}{3}{x}^3$

Supplement: Replacement formula

• $u(x{)}^{v(x)}\to e^{v\left(x\right)lnu\left(x\right)}$
• $ln(\dots \dots )\to ln(1+△)\sim △$
• $(\dots \dots )-1\to \{\begin{array}{l}{e}^{△}-1\sim △\\ (1+△{)}^a-1\sim a△\end{array}$

（2）$\frac{\infty }{\infty }$

$\frac{\infty }{\infty }$: $\{\begin{array}{l}\frac{0}{0}\\ Lópida\text{'}sLaw:\_\frac{0}{0}or\frac{\infty }{\infty }\\ \underset{x\to +\infty }{\lim }\frac{a_m{x}^m+\dots \dots +a_{1}x+a_0}{b_n{x}^n+\dots \dots +b_{1}x+b_0}=\{\begin{array}{l}0，m<n\\ \infty ，m>n\\ \frac{a_m}{b_n}，m=n\end{array}\end{array}$

Supplement: infinity

• To x $\to +\infty$时， $l{n}^{\alpha }x\ll x^b\ll a^x$

• Ton $\to \infty$时，$l{n}^{\alpha }n\ll n^b\ll a^n\ll n!\ll n^n$

• infinite amount $\to$ Unbounded variable, but not vice versa [$(-1{)}^n$】

（3）$1^{\infty }$

$1^{\infty }$ : Identity deformation,$(1+△{)}^{\frac{1}{△}}\sim e$

（4）${\infty }^0$、$0^{\infty }$

$\{\begin{array}{l}{\infty }^0\\ 0^{\infty }\end{array}\to u(x{)}^{v(x)}\to e^{v\left(x\right)lnu\left(x\right)}$

（5） $0\times \infty$

$0\times \infty$：$\{\begin{array}{l}\frac{0}{\frac{1}{\infty }}\to \frac{0}{0}\\ \\ \frac{\infty }{\frac{1}{0}}\to \frac{\infty }{\infty }\end{array}$

（6）$\infty -\infty$

$\infty -\infty ：\to 0\times \infty$ Same as above

Supplement: Substitution Principle

• This value is a continuous point in the function interval [ $\underset{x\to x_0}{\lim }f(x)=f(x_0)$】
• Partial factor limit $\exists$即可拆分【$\exists \underset{x\to x_0}{\lim }f\left(x\right)=A，\underset{x\to x_0}{\lim }[f(x)\pm g(x)]=A+\underset{x\to x_0}{\lim }g\left(x\right)$ (in questions involving unknown numbers, the limit should be removed first$\exists$）
• The addition and subtraction relationships can be exchanged under certain conditions
  [ $a\sim a_1,b\sim b_1,and\{\begin{array}{l}\lim =\frac{a_1}{b_1}=A\ne 1,Then\alpha -b\sim a_1-b_1\\ \lim =\frac{a_1}{b_1}=A\ne -1,Then\alpha +b\sim a_1+b_1\end{array}$】

3. Mean value theorem

(6) Judgment of continuity and properties of continuous functions

1. Determine continuity

1) Definition

设$y=f\left(x\right)$ at point$x_0$There is a definition in a certain decentered neighborhood, then $\{\begin{array}{l}\underset{△x\to 0}{\lim }△y=\underset{△x\to 0}{\lim }[f(x_0+△x)-f(x_0)]\\ \underset{△x\to 0}{\lim }f(x)=f(x_0)\end{array}$

2) Theorem

① $f\left(x\right)andg\left(x\right)$ are at point$x_0$are continuous, then $f(x)\pm g(x)，f(x)g(x)，\frac{f(x)}{g(x)}(g(x)\ne 0)$ is continuous at the point

②Suppose function $u=\varphi \left(x\right)$ at point$x_0$is continuous at , and $\varphi (x_0)=u_0$， function $y=f\left(u\right)$ point$u_0$is continuous at , then $y=f\left[\varphi \right(x\left)\right]$ at$x=x_0$Continuous everywhere

③Basic elementary functions are continuous within their domain

④Elementary functions are continuous within their definition interval

2. Properties of continuous functions

设$f(x)\in C[a,b]$

1） （m，M）$f\left(x\right)$ has min on [a,b], MAX
2)(bounded) $\exists k>0,so\; that|f(x)|\le k$
3)(zero point) $Andf\left(a\right)f\left(b\right)<0，\exists c\in [a,b],so\; thatf\left(c\right)=0$
4)(intermediate value) $\forall \eta \in [m,M],then\exists \xi \in [a,b],so\; thatf\left(\xi \right)=the$

(7) Types of discontinuities

Category 1: $f(a-0)=f(a+0)$ all have values

$\{\begin{array}{l}Canremovediscontinuities:f(a\_-0)=f(a+0)\ne f(a)\\ Jumpdiscontinuitypoint:f(a-0)\ne f(a+0)\end{array}$

The second category: $f(a-0)、f(a+0)$ At least one does not exist

Before judging the discontinuity point (that is, before finding the limit), deform if you can, and don’t rush to substitute.