Sequences

1. Sequences and limits.

Definition 1.1 A sequence of real numbers is a function $f:N\to R,f\left( n \right)={{a}_{n}}$.

Definition 1.2 The sequence $\left( {{a}_{n}} \right)n\ge 0$ is called increasing (respectively decreasing) if ${{a}_{n}}\le {{a}_{n+1}},\forall n\in N$ (respectively ${{a}_{n}}\ge {{a}_{n+1}},\forall n\in N$). Increasing sequences and decreasing sequences are called monotonic sequences.

Definition 1.3 The sequence $\left( {{a}_{n}} \right)n\ge 0$ is bounded if and only if $\exists M>0$ such that $\left| {{a}_{n}} \right|\le M,\forall n\in N$.

Notation: ${{\left( {{a}_{n}} \right)}_{n\ge 0}}, {{a}_{n}}\in R,\text{ }\bar{R}=R\cup \left\{ -\infty ,+\infty \right\}$;

Definition 1.5 The sequence is convergent, $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a,\text{ }a\in R$, if $\forall \varepsilon >0, \exists {{N}_{\varepsilon }}\in N$ such that $\forall n>{{N}_{\varepsilon }}, \left| {{a}_{n}}-a \right|<\varepsilon$.

Definition 1.6 $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\infty$ if $(\forall )\varepsilon >0,\text{ (}\exists ) {{N}_{\varepsilon }}\in N$ such that ${{a}_{n}}>\varepsilon ,\forall n>{{N}_{\varepsilon }}$.

Definition 1.7 $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}_{n}}=-\infty$ if $\forall \varepsilon >0,\exists {{N}_{\varepsilon }}\in N$ such that ${{a}_{n}}<-\varepsilon , \forall n>{{N}_{\varepsilon }}$

If $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\pm \infty$, then the sequence is divergent.

A sequence ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ can be defined in several ways:

• descriptively (e.g. 1,3,5,7,9,…);

• by a formula for the general term (e.g. ${{b}_{n}}={{n}^{2}}-n+1$);

• by a recurrence formula (e.g. ${{b}_{1}}=1$, ${{b}_{2}}=2$, ${{b}_{n+2}}={{b}_{n}}+{{b}_{n+1}}$).

2. Sufficient criteria for convergence, or for a sequence to have a limit.

1. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=0, {{b}_{n}}\ge 0$ and $\left| {{a}_{n}}-a \right|\le {{b}_{n}}$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a$;

2. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=\infty$ and ${{a}_{n}}\ge {{b}_{n}}$, then $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}_{n}}=+\infty$

3. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=-\infty$ and ${{a}_{n}}\le {{b}_{n}}$, then $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}_{n}}=-\infty$;

4. every monotonic bounded sequence is convergent (Weierstrass's criterion);

5. if ${{b}_{n}}\le {{a}_{n}}\le {{c}_{n}}$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=\underset{n\to \infty }{\mathop{\lim }}\,{{c}_{n}}=a$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a$;

6. Stolz's criterion:

- if ${{\left( {{b}_{n}} \right)}_{n\ge 0}}$ is increasing, $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=+\infty$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}-{{a}_{n}}}{{{b}_{n+1}}-{{b}_{n}}}$ exists, then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}=\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}-{{a}_{n}}}{{{b}_{n+1}}-{{b}_{n}}}$

- if ${{\left( {{a}_{n}} \right)}_{n\ge 0}}$, ${{a}_{n}}>0$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}}{{{a}_{n}}}$ exists, then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{{{a}_{n}}}=\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}}{{{a}_{n}}}$ (Cesàro).

- if ${{\left( {{b}_{n}} \right)}_{n\ge 0}}$ is decreasing, $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=0$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}-{{a}_{n}}}{{{b}_{n+1}}-{{b}_{n}}}$ exists, then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}=\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}-{{a}_{n}}}{{{b}_{n+1}}-{{b}_{n}}}$.

7. The ratio test.

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ be a sequence of strictly positive numbers. Suppose $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n+1}}}{{{a}_{n}}}=l$ exists:

• if $l<1\Rightarrow$ $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=0$,

• if $l\in (1,\infty ]\Rightarrow$ $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\infty$.

3. Operations with convergent sequences

$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a, \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=b\text{ }a, b\in \bar{R}$

1.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{n}}+{{b}_{n}} \right)=a+b,\text{ }\underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{n}}\cdot {{b}_{n}} \right)=a\cdot b$;

2.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\alpha \cdot {{a}_{n}}=\alpha \cdot a,\text{ }\alpha \in R$;

3.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{n}}}{{{b}_{n}}}=\frac{a}{b}$ (if $b\ne 0$).

4. Operations with sequences that have a limit.

$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a, \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=b\text{ }a,b\in \bar{R}$

1. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\infty$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=b,\text{ }b\in R$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{n}}+{{b}_{n}} \right)=+\infty$, $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\cdot {{b}_{n}}=\left\{ \begin{aligned} & +\infty ,\, \,dac\breve{a}\,\,b>0 \\ & -\infty , \,\,dac\breve{a}\,\,b<0 \\ \end{aligned} \right.,\text{ }\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{a}_{n}}}=0$;

2. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=+\infty$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{n}}+{{b}_{n}} \right)=+\infty ,$ $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\cdot {{b}_{n}}=+\infty$;

3. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=-\infty$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=b,\text{ }\,\,b\in R$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{n}}+{{b}_{n}} \right)=-\infty$, $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\cdot {{b}_{n}}=\left\{ \begin{aligned} & -\infty \,, \,dac\breve{a}\,\,b>0 \\ & +\infty ,\,\text{ }\,dac\breve{a}\,\,b<0 \\ \end{aligned} \right.;$

4. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=\underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=-\infty$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{n}}+{{b}_{n}} \right)=-\infty ,$ $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\cdot {{b}_{n}}=+\infty$;

5. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=+\infty$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{b}_{n}}=-\infty$, then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\cdot {{b}_{n}}=-\infty$;

6. if $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=0$ then $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{a}_{n}}}=\infty$ if ${{a}_{n}}>0$ and $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{a}_{n}}}=-\infty$ if ${{a}_{n}}<0$.

6. Indeterminate forms

$\frac{\pm \infty }{\pm \infty }, \frac{0}{0}, \infty -\infty , \infty \cdot 0, {{1}^{\infty }}, {{\infty }^{0}}, {{0}^{0}}$

6. Standard sequences

1.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{q}^{n}}=\left\{ \begin{aligned} & 0,\,dac\breve{a}\,\,-1<q<1 \\ & 1,\,dac\breve{a}\,\,q=1 \\ & +\infty ,\,\,dac\breve{a}\,\,q>1 \\ & nu\,\,exist\breve{a},\,\,dac\breve{a}\,\,q\le -1 \\ \end{aligned} \right.;$

2.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( {{a}_{0}}{{n}^{k}}+{{a}_{1}}{{n}^{k-1}}+...+{{a}_{k-1}}n+{{a}_{k}} \right)=\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{0}}{{n}^{k}}=\left\{ \begin{aligned} & +\infty ,\,\,dac\breve{a}\,\,{{a}_{0}}>0 \\ & -\infty ,\,\,dac\breve{a}\,\,{{a}_{0}}<0 \\ \end{aligned} \right.;$

3.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}_{0}}{{n}^{k}}+{{a}_{1}}{{n}^{k-1}}+...+{{a}_{k-1}}n+{{a}_{k}}}{{{b}_{0}}{{n}^{p}}+{{b}_{1}}{{n}^{p-1}}+...+{{b}_{p-1}}n+{{b}_{p}}}=\left\{ \begin{aligned} & 0,\, dac\breve{a}\,k<p \\ & +\infty ,\, dac\breve{a}\,k>p\, i\, {{a}_{0}}{{p}_{0}}>0 \\ & -\infty , \,dac\breve{a}\,k>p \,i\, {{a}_{0}}{{p}_{0}}<0 \\ & \frac{{{a}_{0}}}{{{b}_{0}}}, \,dac\breve{a}\,k=p \\ \end{aligned} \right.$ ;

4.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( 1+q+{{q}^{2}}+...+{{q}^{n}} \right)=\frac{1}{1-q},\,\text{ }dac\breve{a}\left| q \right|<1$;

5.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n} \right)=+\infty$;

6.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{a}=1, \,\forall a>0$;

7.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{{{1}^{p}}+{{2}^{p}}+...+{{n}^{p}}}=1, \,\forall p\ge 1$;

8.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{n} \right)}^{n}}=e$;

9.$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( 1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!} \right)=e$;

10. $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{n}=1$;

11. $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{a}=1$;

12. $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{\left( 1+\frac{1}{n} \right)}^{p}}-1}{\frac{1}{n}}=p$;

13. $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{a}^{n}}}{n!}=0$, $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{{{n}^{n}}}{n!}=0$;

14. $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\frac{\ln \left( 1+\frac{1}{n} \right)}{\frac{1}{n}}=1$;

15. $\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,{{({{u}_{n}})}^{{{v}_{n}}}}=\underset{n\to \infty }{\mathop{\lim }}\,{{e}^{{{v}_{n}}({{u}_{n}}-1)}}$.

Limits of functions

Notation: $f:D\to R,D\subset R,\alpha$ – an accumulation point of D ;

1. Definitions of the limit

Definition 1.1 $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=l,l\in \bar{R}$, if for every neighbourhood V of l there is a neighbourhood U of $\alpha$ such that $\forall x\in D\cap U,x\ne \alpha$ implies $f\left( x \right)\in V$;

Definition 1.2 $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=l, l\in \bar{R}$, if for every sequence ${{\left( {{x}_{n}} \right)}_{n\ge 0}}$, ${{x}_{n}}\in D\backslash \left\{ \alpha \right\}$, with $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{x}_{n}}=\alpha$ it follows that $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,f\left( {{x}_{n}} \right)=l$ (the sequential criterion);

Definition 1.3 $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=l,\, l\in \bar{R}$, if $\forall \varepsilon >0, \exists {{\delta }_{\varepsilon }}>0$ such that $\forall x\in D\backslash \left\{ \alpha \right\}$ and $\left| x-\alpha \right|<{{\delta }_{\varepsilon }}$ imply $\left| f\left( x \right)-l \right|<\varepsilon$;

Definition 1.4 $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=l$, if ${{l}_{s}}={{l}_{d}}=l$, where $\displaystyle {{l}_{s}}=\underset{\begin{smallmatrix} x\to \infty \\ x<\alpha \end{smallmatrix}}{\mathop{\lim }}\,f\left( x \right)$ and $\displaystyle {{l}_{d}}=\underset{\begin{smallmatrix} x\to \infty \\ x>\alpha \end{smallmatrix}}{\mathop{\lim }}\,f\left( x \right)$.

2. Operations with limits of functions

$f:D\to R,g:D\to R,\alpha$ – an accumulation point of D, $\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,f\left( x \right)={{l}_{1}},\text{ }\underset{x\to \alpha }{\mathop{\lim }}\,g\left( x \right)={{l}_{2}}, {{l}_{1}}, {{l}_{2}}\in R$;

1.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)={{l}_{1}}+{{l}_{2}}$;

2.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,f\left( x \right)\cdot g\left( x \right)={{l}_{1}}\cdot {{l}_{2}}$;

3.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,af\left( x \right)=a\cdot {{l}_{1}},\forall a\in R$

4. if $\displaystyle {{l}_{2}}\ne 0,\text{ }\underset{x\to \infty }{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{{{l}_{1}}}{{{l}_{2}}}$

3. Standard limits

1.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\left( {{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+...+{{a}_{n}} \right)={{a}_{0}}{{\alpha }^{n}}+{{a}_{1}}{{\alpha }^{n-1}}+...+{{a}_{n}}$;

$\displaystyle \underset{x\to \pm \infty }{\mathop{\lim }}\,\left( {{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+...+{{a}_{n}} \right)=\underset{x\to \infty }{\mathop{\lim }}\,{{a}_{0}}{{x}^{n}}$;

2.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\frac{{{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+...+{{a}_{n}}}{{{b}_{0}}{{x}^{m}}+{{b}_{1}}{{x}^{m-1}}+...+{{b}_{m}}}=\frac{{{a}_{0}}{{\alpha }^{n}}+{{a}_{1}}{{\alpha }^{n-1}}+...+{{a}_{n}}}{{{b}_{0}}{{\alpha }^{m}}+{{b}_{1}}{{\alpha }^{m-1}}+...+{{b}_{m}}}$

$\displaystyle \underset{x\to \pm \infty }{\mathop{\lim }}\,\frac{{{a}_{0}}{{x}^{n}}+{{a}_{1}}{{x}^{n-1}}+...+{{a}_{n}}}{{{b}_{0}}{{x}^{m}}+{{b}_{1}}{{x}^{m-1}}+...+{{b}_{m}}}=\underset{x\to \pm \infty }{\mathop{\lim }}\,\frac{{{a}_{0}}{{x}^{n}}}{{{b}_{0}}{{x}^{m}}}$;

3.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\sqrt[n]{x}=\sqrt[n]{\alpha },\alpha \in {{R}_{+}},n\in N,n\ge 2$;

$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\sqrt[n]{x}=\infty ,\text{ }\underset{x\to -\infty }{\mathop{\lim }}\,\sqrt[2n+1]{x}=-\infty$;

4.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,{{a}^{x}}={{a}^{\alpha }},\alpha \in R,a\in R_{+}^{*}\backslash \left\{ 1 \right\}$;

$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}^{x}}=\infty ,\underset{x\to \infty }{\mathop{\lim }}\,{{a}^{x}}=0,\,\text{ }dac\breve{a}\,\,\,\,a>1$;

$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{a}^{x}}=0,\underset{x\to \infty }{\mathop{\lim }}\,{{a}^{x}}=\infty ,\text{ }\,dac\breve{a}\,\,\,0<a<1$

5.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,{{\log }_{a}}x={{\log }_{a}}\alpha , \alpha >0$ finite, $a\in R_{+}^{*}\backslash \left\{ 1 \right\}$

$\displaystyle \underset{\begin{smallmatrix} x\to 0 \\ x>0 \end{smallmatrix}}{\mathop{\lim }}\,{{\log }_{a}}x=-\infty$ and $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{\log }_{a}}x=+\infty$ if $a>1$;

$\displaystyle \underset{\begin{smallmatrix} x\to 0 \\ x>0 \end{smallmatrix}}{\mathop{\lim }}\,{{\log }_{a}}x=+\infty$ and $\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,{{\log }_{a}}x=-\infty$ if

6.$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\sin x=\sin \alpha$, $\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\cos x=\cos \alpha$;

$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,tgx=tg\alpha ,\text{ }\alpha \in \frac{\pi }{2}+\pi Z$;

$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,ctgx=ctg\alpha ,\text{ }\alpha \notin \pi Z$;

$\displaystyle \underset{\begin{smallmatrix} x\to \frac{\pi }{2} \\ x<\frac{\pi }{2} \end{smallmatrix}}{\mathop{\lim }}\,tgx=\infty ,\text{ }\underset{\begin{smallmatrix} x\to \frac{\pi }{2} \\ x>\frac{\pi }{2} \end{smallmatrix}}{\mathop{\lim }}\,tgx=-\infty$;

7.$\displaystyle \underset{\begin{smallmatrix} x\to \frac{\pi }{2} \\ x<\frac{\pi }{2} \end{smallmatrix}}{\mathop{\lim c}}\,tgx=\infty ,\text{ }\underset{\begin{smallmatrix} x\to 0 \\ x>0 \end{smallmatrix}}{\mathop{\lim c}}\,tgx=-\infty$;

$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\,\arcsin x=\arcsin \alpha ,\alpha \in \left[ -1,1 \right]$;

$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\,\operatorname{ar}\cos x=\operatorname{ar}\cos \alpha ,\alpha \in \left[ -1,1 \right]$;

$\displaystyle \underset{x\to \alpha }{\mathop{\lim }}\,\,arctgx=arctg\alpha ,\alpha \in R$;

;

$\displaystyle \underset{x\to -\infty }{\mathop{\lim }}\,arctgx=-\frac{\pi }{2},\text{ }\underset{x\to \infty }{\mathop{\lim }}\,arctgx=\frac{\pi }{2}$;

$\displaystyle \underset{x\to -\infty }{\mathop{\lim }}\,arcctgx=\pi ,\text{ }\underset{x\to \infty }{\mathop{\lim }}\,arcctgx=0$;

8.$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}=1;\text{ }\underset{x\to 0}{\mathop{\lim }}\,\frac{tgx}{x}=1;$

$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{\arcsin x}{x}=1;\text{ }\underset{x\to 0}{\mathop{\lim }}\,\frac{arctgx}{x}=1;$

9.$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{a}^{x}}}=0,\text{ }\forall n\in Z, a>1$;

10.$\displaystyle \underset{x\to \pm \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{x} \right)}^{x}}=e,\text{ }\underset{x\to 0}{\mathop{\lim }}\,{{\left( 1+x \right)}^{\frac{1}{x}}}=e$;

11.$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{\ln \left( 1+x \right)}{x}=1$;

12.$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-1}{x}=\ln a,a>0$

Graph of the function

$f(x)=\frac{{{\left( 1+x \right)}^{r}}-1}{x}=r$ for r=3

13.$\displaystyle \underset{x\to 0}{\mathop{\lim }}\,\frac{{{\left( 1+x \right)}^{r}}-1}{x}=r, \forall r\in R$.