Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(description)
(limits)
Line 5: Line 5:
 
|-  
 
|-  
 
| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 7}}
 
| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 7}}
−
==<span style="font-size:20px; color: blue;">Combinatorics</span>==
+
==<span style="font-size:20px; color: blue;">Limits</span>==
−
This section cover combinatorics, and some binomial/multinomial facts.
+
This section covers limits and some other precalculus topics.
−
<!-- will fill in later! -->
+
===Definition===
−
===Permutations===
 
−
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>.
 
  
−
Also, <math>0!=1</math>.
+
*<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>.
  
−
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>
+
*<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>.
  
−
===Combinations===
+
*<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>.
−
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.
 
  
−
===Binomials and Multinomials===
+
*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>.
−
*Binomial Theorem: <math>(x+y)^n=\sum_{r=0}^{n}x^{n-r}y^r</math>
 
−
*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>
 
−
*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>.
 
  
−
[[User:Temperal/The Problem Solver's Resource6|Back to page 6]] | [[User:Temperal/The Problem Solver's Resource8|Continue to page 8]]
+
===Theorems and Properties===
 +
 
 +
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>.
 +
 
 +
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>
 +
 
 +
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>.
 +
 
 +
 
 +
[[User:Temperal/The Problem Solver's Resource4|Back to page 4]] | [[User:Temperal/The Problem Solver's Resource6|Continue to page 6]]
 
|}<br /><br />
 
|}<br /><br />

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


Back to page 4 | Continue to page 6



Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User:Temperal/The_Problem_Solver%27s_Resource7&oldid=17462"