• A limit of a sum is the sum of the limits
  
  
  
  
• A limit of a product is the product of the limits
  
  
  
  
• The limit of a quotient is the quotient of the limits
  
  
  
  
• The limit of a constant is a constant
  
  
  
  
• The limit of a constant coefficient of a function is the limit of the function multiplied by the coefficient
  
  
  
  
• If is continuous at and then
  
  
  
  
  
  • We can derive other limits.
    
    Given where is constant and belongs to the set of rational numbers , then
    
    
    
  
• If for all then if and approach some limit , , then must approach the same limit
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• Left and Right Limits
• Another way to address a limit at a point is to address the limit defined by the behavior of the function to the left of the point and to the right of the point
  
  
  • CONSIDER: The limits of the heaviside function at
    
    
    
    
    
    The limit defined by the behavior of to the left of is given by
    
    
    
    The limit defined by the behavior of to the right of is given by
    
    
    
    The limit defined by the overall behavior of at is given by the original definition of the limit. As such it does not exist
    
    
• Left and right limits allow us to predict the behavior of discontinuous functions, or functions that are continuous on their domain, but contain asymptotes
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  Limits at Infinity
• Limits at infinity for a continuous defined on , is similar to the limits of a sequence as it approaches infinity
  
  
• Similarly the limit at negative infinity () for a continuous function defined on is simply the reversal of the inequality
  
  "There exists a value such that whenever then "
  
  
• A graphical representation is the graph of
  
  
  
  
  
  We know that as then
  
  We can write in the form thus,
  
  The formal epsilon definition simply states for an arbitrarily chosen , there exists a value , such that whenever is greater than , then the value of are always less than the range of values created from
  
  
• The value of is dependant on our choice of
  
  
  
• Continuity

  • A function if the following holds
    
    
  DEFINITION
  For inputs and in order for to be so close to so that (for an arbitrarily chosen ), it suffices that and are soo close that (where is dependant on the choice of )
  • CONSIDER: The Heaviside function , is it possible for any chosen to force two values and to be close such that ?
    
    
          
    
    
    If and are the same sign (the same sign property ) then no matter how close we choose and , they will always satisfy the condition
    
    
    
    
    
    If and are opposite signs then the condition is dependant on our choice of . If epsilon is chosen such that then the condition holds regardless of how we choose and
    
    
    
    
    
    However, if then the condition does not hold if and are opposite signs.
    
    
  • The definition of a continuous function is essentially a limit
    
    
  DEFINITION
  A function is continuous at if Implying that
  • A function is continuous on an interval if it is continuous at each point on the interval

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By Muhammad Atif Saeed on 21:23. Filed under feature , Mathandstat . Follow any responses to the RSS 2.0. Leave a response