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

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==<span style="font-size:20px; color: blue;">Combinatorics</span>==
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==<span style="font-size:20px; color: blue;">Limits</span>==
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This section cover combinatorics, and some binomial/multinomial facts.
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This section covers limits and some other precalculus topics.
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===Definition===
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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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===Binomials and Multinomials===
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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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*Binomial Theorem: <math>(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>\sum_{i=1}^{s}r_i</math> so that <math>\sum_{i=1}^{s}r_i=n</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]]
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===Theorems and Properties===
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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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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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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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[[User:Temperal/The Problem Solver's Resource4|Back to page 4]] | [[User:Temperal/The Problem Solver's Resource6|Continue to page 6]]
 
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Revision as of 12:56, 6 October 2007



The Problem Solver's Resource
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 $n$.

Theorems and Properties

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 $0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon$.

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

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

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 $\lim_{x\to S}f(x)=L$.


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