# Difference between revisions of "User:Temperal/The Problem Solver's Resource5"

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*If <math>\displaystyle\lim_{x\to n}f(x)=f(n)</math>, then <math>f(x)</math> is said to be continuous in <math>n</math>. | *If <math>\displaystyle\lim_{x\to n}f(x)=f(n)</math>, then <math>f(x)</math> is said to be continuous in <math>n</math>. | ||

− | ==Theorems and Properties== | + | ===Theorems and Properties=== |

− | The statement <math> | + | 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: | Let <math>f</math> and <math>g</math> be real functions. Then: | ||

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*<math>\lim(\frac{f}{g})(x)=\frac{\lim f(x)}{\lim g(x)}</math> | *<math>\lim(\frac{f}{g})(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> | + | 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]] | [[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 14:25, 30 September 2007

## LimitsThis 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 .
## Theorems and PropertiesThe statement is equivalent to: given a positive number , there is a positive number such that . Let and be real functions. Then: Suppose is between and for all in the neighborhood of . If and approach some common limit L as approaches , then . |