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Difference between revisions of "User:Temperal/The Problem Solver's Resource7"

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{| style='background:lime;border-width: 5px;border-color: limegreen;border-style: outset;opacity: 0.8;width:840px;height:300px;position:relative;top:10px;'
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==<span style="font-size:20px; color: blue;">Limits</span>==
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|+ <span style="background:aqua; border:1px solid black; opacity: 0.6;font-size:30px;position:relative;bottom:8px;border-width: 5px;border-color:blue;border-style: groove;position:absolute;top:50px;right:155px;width:820px;height:40px;padding:5px;">The Problem Solver's Resource</span>
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This section covers limits and some other precalculus topics.
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===Definition===
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| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 7}}
 
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==<span style="font-size:20px; color: blue;">Combinatorics</span>==
 
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===Permutations===
 
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The factorial of a number <math>n</math> is <math>n(n-1)(n-2)...(1)</math> or also as <math>\prod_{a=0}^{n-1}(n-a)</math>,and is denoted by <math>n!</math>.
 
  
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Also, <math>0!=1</math>.
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*<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>.
  
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The number of ways of arranging <math>n</math> distinct objects in a straight line is <math>n!</math>. This is also known as a permutation, and can be notated <math>\,_{n}P_{r}</math>
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*<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>.
  
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===Combinations===
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*<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>.
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The number of ways of choosing <math>n</math> objects from a set of <math>r</math> objects is <math>\frac{n!}{r!(n-r)!}</math>, which is notated as either <math>\,_{n}C_{r}</math> or <math>\binom{n}{r}</math>. (The latter notation is also known as taking the binomial coefficient.
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*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>.
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===Properties===
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Let <math>f</math> and <math>g</math> be real functions. Then:
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*<math>\lim(f+g)(x)=\lim f(x)+\lim g(x)</math>
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*<math>\lim(f-g)(x)=\lim f(x)-\lim g(x)</math>
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*<math>\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)</math>
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*<math>\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}</math>
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===Squeeze Play Theorem (or Sandwich Theorem)===
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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>.
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===Diverging-Converging Theorem===
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A series <math>\sum_{i=0}^{\infty}S_i</math> converges iff <math>\lim S_i=0</math>.
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===Focus Theorem===
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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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===Binomials and Multinomials===
 
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*Binomial Theorem: <math>\displaystyle (x+y)^n=\sum_{r=0}^{n}x^{n-r}y^r</math>
 
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*Multinomial Coefficients: The number of ways of ordering <math>n</math> objects when <math>r_1</math> of them are of one type, <math>r_2</math> of them are of a second type, ... and <math>r_s</math> of them of another type is <math>\frac{n!}{r_1!r_2!...r_s!}</math>
 
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*Multinomial Theorem: <math>(x_1+x_2+x_3...+x_s)^n=\sum \frac{n!}{r_1!r_2!...r_s!} x_1+x_2+x_3...+x_s</math>. The summation is taken over all sums <math>\displaystyle\sum_{i=1}^{s}r_i</math> so that <math>\displaystyle\sum_{i=1}^{s}r_i=n</math>.
 
  
 
[[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

  • $\lim_{x\to n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$.
  • $\lim_{x\uparrow n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$ from values of $x$ less than $n$.
  • $\lim_{x\downarrow n}f(x)$ is the value that $f(x)$ approaches as $x$ approaches $n$ from values of $x$ more than $n$.
  • If $\lim_{x\to n}f(x)=f(n)$, then $f(x)$ is said to be continuous in $n$.

Properties

Let $f$ and $g$ be real functions. Then:

  • $\lim(f+g)(x)=\lim f(x)+\lim g(x)$
  • $\lim(f-g)(x)=\lim f(x)-\lim g(x)$
  • $\lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)$
  • $\lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}$

Squeeze Play Theorem (or Sandwich Theorem)

Suppose $f(x)$ is between $g(x)$ and $h(x)$ for all $x$ in the neighborhood of $S$. If $g$ and $h$ approach some common limit L as $x$ approaches $S$, then $\lim_{x\to S}f(x)=L$.


Diverging-Converging Theorem

A series $\sum_{i=0}^{\infty}S_i$ converges iff $\lim S_i=0$.

Focus Theorem

The statement $\lim_{x\to n}f(x)=L$ is equivalent to: given a positive number $\epsilon$, there is a positive number $\gamma$ such that $0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon$.


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