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limit rules of functions
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(Theorem)
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Theorem 1 Let $f$ and $g$ be two real 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
- $\lim_{x\to x_0}[f(x)\!\pm\!g(x)] = \lim_{x\to x_0}f(x)\pm\lim_{x\to x_0}g(x),$
- $\lim_{x\to x_0}f(x)g(x) = \lim_{x\to x_0}f(x)\cdot\lim_{x\to x_0}g(x),$
- $\lim_{x\to x_0}\frac{f(x)}{g(x)} = \frac{\lim_{x\to x_0}f(x)}{\lim_{x\to x_0}g(x)},$
- $\lim_{x\to x_0}c = c \quad\mathrm{where}\,\,c\,\,\mathrm{is\,\,a\,\,constant}.$
These rules are used in limit calculations and in proving the corresponding differentiation rules (sum rule, product rule etc.).
In theorem 1, the domains of $f$ and $g$ could be any topological space (not necessarily $\mathbb{R}$ or $\mathbb{C}$ ).
There are similar limit rules of sequences.
As well, one often needs the
Theorem 2 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 $$\lim_{x\to x_0}g(f(x)) = g(\lim_{x\to x_0}f(x)).$$
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"limit rules of functions" is owned by pahio.
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Cross-references: point, continuous at, sequences, similar, topological space, domains, product rule, sum rule, differentiation, limits, complex functions
There are 7 references to this entry.
This is version 15 of limit rules of functions, born on 2004-11-25, modified 2008-01-26.
Object id is 6528, canonical name is LimitRulesOfFunctions.
Accessed 21865 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) | | | 30A99 (Functions of a complex variable :: General properties :: Miscellaneous) |
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Pending Errata and Addenda
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