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[parent] limit rules of functions (Theorem)
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
  1. $ \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),$
  2. $ \lim_{x\to x_0}f(x)g(x) = \lim_{x\to x_0}f(x)\cdot\lim_{x\to x_0}g(x),$
  3. $ \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)},$
  4. $ \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
$\displaystyle \lim_{x\to x_0}g(f(x)) = g(\lim_{x\to x_0}f(x)).$



"limit rules of functions" is owned by pahio.
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See Also: growth of exponential function, improper limits, derivatives of sine and cosine, list of common limits, limit examples, sum and product and quotient of functions


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proof of limit rule of product (Proof) 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 12443 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
Pending Errata and Addenda
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limit rule of compound function by pahio on 2008-01-26 14:10:58
Hi activists,
The new member urscheljd has asked the proof of the Theorem 2 in http://planetmath.org/encyclopedia/LimitRulesOfFunctions.html
Is there someone member who would want to add such a proof in PM?
Regards,
Jussi
[ reply | up ]
limit rule of compound function by pahio on 2008-01-26 14:10:47
Hi activists,
The new member urscheljd has asked the proof of the Theorem 2 in http://planetmath.org/encyclopedia/LimitRulesOfFunctions.html
Is there someone member who would want to add such a proof in PM?
Regards,
Jussi
[ reply | up ]
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