Limits of functions

1) $f(x)={{e}^{-{{x}^{2}}}}$

• maximal domain of definition: $\mathbb{R}$;

• aperiodic function;

• the graph does not meet the Ox axis*;*

• intersection with the Oy axis: (0,1);

• the function is even;

• the Oy axis is a horizontal asymptote;

• $\displaystyle \underset{x\to \pm \infty }{\mathop{\lim} }\,f(x)=0$ is continuous on $\mathbb{R}$;

• also known as the "Gaussian bell curve".

2) $f(x)=\frac{\sin x}{x}$

• maximal domain of definition: $\mathbb{R}$;

• periodic function, with fundamental period 2π ;

• the graph meets the Oy axis at (0,1), and the Ox axis at $(k\pi ,0); k\in \mathbb{Z}\backslash \{0\}$ ;

• the function is even;

• $\displaystyle \underset{x\to 0}{\mathop{\lim} }\,\frac{\sin (x)}{x}=1$ $\displaystyle \underset{x\to \pm \infty }{\mathop{\lim} }\,f(x)=0$ has no asymptotes;

• it is continuous on $\mathbb{R}$;

• known as the "damped sine".

3) $f(x)=\frac{\cos x}{x}$

• maximal domain of definition: $\mathbb{R}\backslash \{0\}$;

• periodic function, with no fundamental period;

• the graph does not meet the Oy axis*;*

• the graph meets the Ox axis at the points $\left( \frac{(2k+1)\pi }{2},0 \right); k\in \mathbb{Z}$

• $\displaystyle \underset{\begin{smallmatrix} x\to 0 \\ x<0 \end{smallmatrix}}{\mathop{\lim} }\,f(x)=-\infty$ $\displaystyle \underset{\begin{smallmatrix} x\to 0 \\ x>0 \end{smallmatrix}}{\mathop{\lim} }\,f(x)=\infty$ the function is odd;

• the Oy axis is a vertical asymptote;

• $\displaystyle \underset{x\to \pm \infty }{\mathop{\lim} }\,f(x)=0$ is continuous on $\mathbb{R}\backslash \{0\}$;

• known as the "damped cosine".

4) $f(x)=\frac{tgx}{x}$

• maximal domain of definition: $\mathbb{R}\backslash \left\{ 0,\frac{(2k+1)\pi }{2} \right\}\text{, }k\in \mathbb{Z}$;

• periodic function, with no fundamental period;

• the graph meets the Oy axis at (0,1);

• the graph meets the Ox axis at the points $(k\pi ,0), k\in \mathbb{Z}\backslash \{0\}$;

• the function is even;

• the lines $x=\frac{(2k+1)\pi }{2}, k\in \mathbb{Z}$ are vertical asymptotes ;

• $\displaystyle \underset{x\to 0}{\mathop{\lim} }\,\frac{tg(x)}{x}=1$ is continuous on $\left( -\frac{\pi }{2},\frac{\pi }{2} \right)$;

• known as the "damped tangent".

5) $f(x)=\frac{ctgx}{x}$

• maximal domain of definition: $\mathbb{R}\backslash \left\{ 0,k\pi \right\}\text{, }k\in \mathbb{Z}\backslash \{0\}$;

• periodic function, with no fundamental period;

• the graph does not meet the Oy axis;

• the graph meets the Ox axis at the points $\left( \frac{(2k+1)\pi }{2},0 \right); k\in \mathbb{Z}$;

• the function is even;

• the lines $x=k\pi , k\in \mathbb{Z}\backslash \{0\}$ are vertical asymptotes ;

• it is continuous on (0, π);

• known as the "damped cotangent".

6)$f(x)=\frac{\arcsin (x)}{x}$

• maximal domain of definition: $\mathbb{R}$;

• aperiodic function;

• the graph does not meet the Ox axis*;*

• intersection with the Oy axis: (0,1);

• the function is even;

• the Oy axis is a horizontal asymptote;

• $\displaystyle \underset{x\to \pm \infty }{\mathop{\lim} }\,f(x)=0$ is continuous on $\mathbb{R}$;

7)$f(x)=\frac{arctg(x)}{x}$

• maximal domain of definition: $\mathbb{R}$;

• aperiodic function;

• the graph does not meet the Ox axis*;*

• intersection with the Oy axis: (0,1);

• the function is even;

• the Oy axis is a horizontal asymptote;

• $\displaystyle \underset{x\to \pm \infty }{\mathop{\lim} }\,f(x)=0$ is continuous on $\mathbb{R}$;

6)$f(x)=\frac{\ln (1+x)}{x}$

• maximal domain of definition: $(-1,\infty )$;

• aperiodic function;

• the graph does not meet the Ox axis*;*

• intersection with the Oy axis: (0,1);

• the Oy axis is a horizontal asymptote;

• $\displaystyle \underset{x\to \pm \infty }{\mathop{\lim} }\,f(x)=0$ is continuous on $\mathbb{R}$;

7)$f(x)=\frac{{{2}^{x}}-1}{x}$

• maximal domain of definition: $\mathbb{R}$;

• aperiodic function;

• the graph does not meet the Ox axis*;*

• intersection with the Oy axis: $(0,\ln 2)$;

• the Oy axis is a horizontal asymptote;

• $\displaystyle \underset{x\to -\infty }{\mathop{\lim} }\,f(x)=0$ is continuous on $\mathbb{R}$;