Ch 1. Limits     Multimedia Engineering Math
Limit of a
Sequence		 Limit of a
Function		 Limit
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Math
 Limit of a Function Case Intro Theory Case Solution  
 Chapter
 1. Limits
 2. Derivatives I
 3. Derivatives II
 4. Mean Value
 5. Curve Sketching
 6. Integrals
 7. Inverse Functions
 8. Integration Tech.
 9. Integrate App.
 10. Parametric Eqs.
 11. Polar Coord.
 12. Series
 
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Author(s):
Hengzhong Wen
Chean Chin Ngo
Meirong Huang
Kurt Gramoll
 
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MATHEMATICS - THEORY
    Function


The quantity relationship of
function y = x2 + x + 1

 

If the quantity of y is determined by another value x, then y is a function of x. If f denotes the function, then the formula y = f(x) indicate y depends on x. Thus y is a dependent variable and x is a independent variable. An example of a function would be:

     y = x2 + x + 1

The diagram on the left shows that the quantity of y which is equal to x2 + x + 1 depends on the value of x.
 

 
  Limit of a Function
   

If for each positive ε, no matter how small, there is a corresponding positive δ such that if

     

then

     

In this situation, the limit of f(x) is A as x approaches a, and it can be written as:

     
     
    Limits Involving Infinity
In this segment, three kind of situations will be discussed.
     

The limit of f(x) is infinity
when x approaches 1
  
  • The limit of f(x) is ∞ as x approaches a if for each number B, no matter how large, there is a corresponding positive δ, such that if

       

then

     f(x) > B

Its notation is

     
     

The limit of f(x) is 1
when x approaches +∞
  
  • The limit of f(x) is A as x approaches ∞ if for each positive ε, no matter how small, there is a corresponding B such that if

       x > B

then

     

Its notation is

     
     

The limit of f(x) is -∞ when x
approaches -∞
  
  • The limit of f(x) is -∞ as x approaches -∞ if for each number B, no matter how large, there is a corresponding C, such that if

       x< -C

then

     f(x) < -B

Its notation is

     
     

Summary
   Summary
  This section introduces limit of a function and three cases involving infinity. When discussing limit of a function, it must be the limit of a specific point.