|
|
|
Revision difference : limit rules of functions |
| Version current |
Version 14 |
| \begin{thmplain} |
\begin{thmplain} |
|
\, Let $f$ and $g$ be two \PMlinkname{real}{RealFunction} or complex functions.\, Suppose that there exist the limits \,$\lim_{x\to x_0}f(x)$\, and\, $\lim_{x\to x_0}g(x)$.\, Then there exist the limits\, $\lim_{x\to x_0}[f(x)\!\pm\!g(x)]$,\, $\lim_{x\to x_0}f(x)g(x)$\, and, if\, $\lim_{x\to x_0}g(x)\neq 0$,\, also\, $\lim_{x\to x_0}f(x)/g(x)$, and
|
\, Let $f$ and $g$ be two \PMlinkname{real}{RealFunction} or complex functions.\, Suppose that there exist the limits\, $\lim_{x\to x_0}f(x)$\, and\, $\lim_{x\to x_0}g(x)$.\, Then there exist the limits\, $\lim_{x\to x_0}[f(x)\!\pm\!g(x)]$,\, $\lim_{x\to x_0}f(x)g(x)$\, and, if\, $\lim_{x\to x_0}g(x)\neq 0$,\, also\, $\lim_{x\to x_0}f(x)/g(x)$, and
|
| \begin{enumerate} |
\begin{enumerate} |
| \item $\lim_{x\to x_0}[f(x)\!\pm\!g(x)] |
\item $\lim_{x\to x_0}[f(x)\!\pm\!g(x)] |
|
\;=\; \lim_{x\to x_0}f(x)\pm\lim_{x\to x_0}g(x),$
|
= \lim_{x\to x_0}f(x)\pm\lim_{x\to x_0}g(x),$
|
|
\item $\lim_{x\to x_0}f(x)g(x) \;=\; \lim_{x\to x_0}f(x)\cdot\lim_{x\to x_0}g(x),$
|
\item $\lim_{x\to x_0}f(x)g(x) = \lim_{x\to x_0}f(x)\cdot\lim_{x\to x_0}g(x),$
|
|
\item $\lim_{x\to x_0}\frac{f(x)}{g(x)} \;=\;
|
\item $\lim_{x\to x_0}\frac{f(x)}{g(x)} =
|
| \frac{\lim_{x\to x_0}f(x)}{\lim_{x\to x_0}g(x)},$ |
\frac{\lim_{x\to x_0}f(x)}{\lim_{x\to x_0}g(x)},$ |
|
\item $\lim_{x\to x_0}c \;=\; c
|
\item $\lim_{x\to x_0}c = c
|
| \quad\mathrm{where}\,\,c\,\,\mathrm{is\,\,a\,\,constant}.$ |
\quad\mathrm{where}\,\,c\,\,\mathrm{is\,\,a\,\,constant}.$ |
| \end{enumerate} |
\end{enumerate} |
| \end{thmplain} |
\end{thmplain} |
|
|
| These rules are used in limit calculations and in proving the corresponding differentiation rules (sum rule, product rule etc.). |
These rules are used in limit calculations and in proving the corresponding differentiation rules (sum rule, product rule etc.). |
|
|
|
In \PMlinkescapetext{theorem} 1, the domains of $f$ and $g$ could be any topological space (not necessarily $\mathbb{R}$ or $\mathbb{C}$).\\
|
In \PMlinkescapetext{theorem} 1, the domains of $f$ and $g$ could be any topological space (not necessarily $\mathbb{R}$ or $\mathbb{C}$). |
|
|
|
There are \PMlinkescapetext{similar} limit rules of \PMlinkname{sequences}{Sequence}.\\
|
There are similar limit rules of sequences.
|
|
|
| As well, one often needs the |
As well, one often needs the |
|
|
| \begin{thmplain} |
\begin{thmplain} |
|
\,If there exists the limit\, $\lim_{x\to x_0}f(x) = a$\, and if $g$ is continuous at the point\, $x = a$, then there exists the limit\, $\lim_{x\to x_0}g(f(x))$, and
|
\,If there exists the limit\, $\lim_{x\to x_0}f(x) = a$\, and if $g$ is continuous in the point\, $x = a$, then there exists the limit\, $\lim_{x\to x_0}g(f(x))$, and
|
|
$$\lim_{x\to x_0}g(f(x)) \;=\; g(\lim_{x\to x_0}f(x)).$$
|
$$\lim_{x\to x_0}g(f(x)) = g(\lim_{x\to x_0}f(x)).$$
|
| \end{thmplain} |
\end{thmplain} |
|
|
|
|
