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Limits
[Definition] [Limits at a Number] [Limits at Infinity] [Limits Links]
Definition
- Formal Definition of a Limit:
- Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement
- means that for each
 there exists a  such that if
 , then 
- In other words...
- If there is a function f defined on an open interval containing c (except possible c), then there is value from both ends (not necessarily the same from both ends) that approaches the x-value.
Limits at a Number
- Properties of Limits
- Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.
 and 
- Scalar Multiple:
- Sum or Difference:
- Product:
- Qutient:
- Power:
- example: evaluate
Limits at Infinity
- For some limits, as
 approaches a x-value from one side, could approach a totally different value from the other side like in the example below:
- Definition:
- Let f be a function that is defined at every real number in some open interval containing c (except possibly at c itself). The statement
- means that for each
 there exists a  such that  whenever  . Similarly, the statement
- means that for each
 there exits a  such that  whenever  .
- In other words...
- Whenever the limit as it approaches a value equals negative infinity from one side and positive infinity from the other side, the limit is infinite.
- Vertical Asymptotes
- If approaches infinity (or negative infinity) as x approaches c from the right or left, then the line
 is a vertical asymptote of the graph of f.
- example: find the vertical asymptote of
 since  , -1 is the vertical asymptote
Limits Links
Calculus Phobe  - limits videos
One-Sided Limits of a Function - definition of a limit
Limits as x Approaches a Constant - limit quiz with detailed solutions
Limits as x Approaches Infinity - limit quiz with detailed solutions
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