Difference between revisions of "User:Temperal/The Problem Solver's Resource7"
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− | + | {{User:Temperal/testtemplate|page 7}} | |
− | + | ==<span style="font-size:20px; color: blue;">Limits</span>== | |
− | + | This section covers limits and some other precalculus topics. | |
− | + | ===Definition=== | |
− | |||
− | ==<span style="font-size:20px; color: blue;"> | ||
− | |||
− | === | ||
− | |||
− | + | *<math>\lim_{x\to n}f(x)</math> is the value that <math>f(x)</math> approaches as <math>x</math> approaches <math>n</math>. | |
− | + | *<math>\lim_{x\uparrow n}f(x)</math> is the value that <math>f(x)</math> approaches as <math>x</math> approaches <math>n</math> from values of <math>x</math> less than <math>n</math>. | |
− | === | + | *<math>\lim_{x\downarrow n}f(x)</math> is the value that <math>f(x)</math> approaches as <math>x</math> approaches <math>n</math> from values of <math>x</math> more than <math>n</math>. |
− | + | ||
+ | *If <math>\lim_{x\to n}f(x)=f(n)</math>, then <math>f(x)</math> is said to be continuous in <math>n</math>. | ||
+ | |||
+ | ===Properties=== | ||
+ | |||
+ | Let <math>f</math> and <math>g</math> be real functions. Then: | ||
+ | *<math>\lim(f+g)(x)=\lim f(x)+\lim g(x)</math> | ||
+ | *<math>\lim(f-g)(x)=\lim f(x)-\lim g(x)</math> | ||
+ | *<math>\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)</math> | ||
+ | *<math>\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}</math> | ||
+ | |||
+ | ===Squeeze Play Theorem (or Sandwich Theorem)=== | ||
+ | Suppose <math>f(x)</math> is between <math>g(x)</math> and <math>h(x)</math> for all <math>x</math> in the neighborhood of <math>S</math>. If <math>g</math> and <math>h</math> approach some common limit L as <math>x</math> approaches <math>S</math>, then <math>\lim_{x\to S}f(x)=L</math>. | ||
+ | |||
+ | |||
+ | ===Diverging-Converging Theorem=== | ||
+ | A series <math>\sum_{i=0}^{\infty}S_i</math> converges iff <math>\lim S_i=0</math>. | ||
+ | |||
+ | ===Focus Theorem=== | ||
+ | The statement <math>\lim_{x\to n}f(x)=L</math> is equivalent to: given a positive number <math>\epsilon</math>, there is a positive number <math>\gamma</math> such that <math>0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon</math>. | ||
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[[User:Temperal/The Problem Solver's Resource6|Back to page 6]] | [[User:Temperal/The Problem Solver's Resource8|Continue to page 8]] | [[User:Temperal/The Problem Solver's Resource6|Back to page 6]] | [[User:Temperal/The Problem Solver's Resource8|Continue to page 8]] | ||
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Latest revision as of 19:19, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 7. |
Limits
This section covers limits and some other precalculus topics.
Definition
is the value that
approaches as
approaches
.
is the value that
approaches as
approaches
from values of
less than
.
is the value that
approaches as
approaches
from values of
more than
.
- If
, then
is said to be continuous in
.
Properties
Let and
be real functions. Then:
Squeeze Play Theorem (or Sandwich Theorem)
Suppose is between
and
for all
in the neighborhood of
. If
and
approach some common limit L as
approaches
, then
.
Diverging-Converging Theorem
A series converges iff
.
Focus Theorem
The statement is equivalent to: given a positive number
, there is a positive number
such that
.