Limit and Continuity

• Definition: Limit at a real number c (Both side limit).
  
  Given > 0, there exists > 0 such that | f ( x ) - L | < whenever   0 < | x - c | < ,
  then the limit of   f ( x ) at x = c is L, and denoted by ,
  otherwise f ( x ) has no limit.
  
  Roughly speaking, means that whenever x approaches to c
  
  from either side of c, the graph point (x, f(x)) approaches to the point (c, L)
  
  on the plane.



• Definition: Continuity for f ( x ) at x = c ( both side continuity ).
  
  Given > 0, there exists > 0, such that | f ( x ) - f ( c ) | < whenever | x - c| < ,
  then f ( x ) is continuous at x = c. ( i.e ),
  otherwise f ( x ) is not continuous at x = c. ( i.e f ( x ) is discontinuous at x = c ).
  
  Roughly speaking, means that whenever x approaches
  
  to c from either side of c, the graph point (x, f(x)) approaches to the
  
  point (c, f(c)) on the plane.



• Remarks:
  
  ( 1 ) The existence of the limit for f ( x ) at x = c is not related to the existence
  
  of the output of f ( x ) at x = c.
  
  ( 2 ) When , f ( x ) is continuous at x = c.
  ( 3 ) Whenever does not exist or ,
  then f ( x ) is not continuous at x = c.
  
  
  • exists but is not equal to f ( c ), then the graph of f ( x ) has a hole at x = c.
  • does not exist, then the graph of f ( x ) has either a jump at x = c or
    f ( x ) has a vertical asymptote at x = c ( will be defined later ).


• Example1: Show that
  Analysis :
  
  ( 1 ) Given > 0, we need to select > 0   to satisfy that   | f ( x ) - L | < whenever   0 < | x - 4 | < .
  
  ( 2 ) Since ( x - 4 ) =   for x > 0, we need to show <   from the condition  
  
  0 < | x - 4 | <   ( i.e ) ( * ),   the factor should be combined with .
  
  By rearranging ( * ),   we acquire .   But, we need .   We can achieve this by
  
  restricting the   x   values close to 4 such that the lower bound of can be obtained.
  
  Therefore, we can obtained a relationship between and . e.g.   let ( i.e ),
  
  then . So, i.e. .
  
  Proof :
  
  Given > 0,   let   whenever 0 < | x - 4 | < ,
  
  .



• Example2: show that
  
  Analysis :
  
  ( 1 ) We need to show     from the condition   0 < | x - 2 | < .   We can rewrite
  
  to .   We know that     since   0 < | x - 2 | < .  
  
  We need an upper bound for | x + 2 | to yield   .   In order to do so, we will restrict the value
  
  of   x   close to 2 as we did in Example1.
  
  ( 2 ) For example, let     ( i.e ), then .   So,   .
  
  Proof :
  
  Given > 0,   let   ,   0 < | x - 2 | <
  
  .



• Example3: show that
  
  ( 1 ) .
  
  ( 2 ) ,
  
      since  
  
  ( 3 )
  
  since ( 2 ) is true and .
  
  Form ( 1 ), ( 2 ), and ( 3 ), we can have the following proof :
  
  Proof :
  
  Given   > 0,   let ,   0 < | x - c | < ,
  
  



• Remarks:
  
  
  • It is easy to see that
    
    ( 1 )   is continuous at x = c > 0 ( positive real number ).
    
    ( 2 )   is continuous at x = c ( any real number ).
    
    ( 3 )   is continuous at x = c ( any real number ).
    
    
  • The selection of     in example 3 is independent of c. But, it seems like
    
    the selection of     in example 1 could be depending on c.


• Definition: Uniform Continuity
  
  Given   > 0,   there exist   > 0   ( independent of c ) such that | f ( x ) - f ( c ) | <
  
  whenever | x - c | < ,   then f ( x ) is uniformly continuous.



• Remarks:
  
  ( 1 ) To show that f ( x ) is uniformly continuous is not a simple problem.
  
  Sometime, it could involve advance concept.
  
  ( 2 ) To show that f ( x ) is not uniformly continuos, we may show that
  
  "There exists > 0,   for any given > 0,   there exist x and c such that | f ( x ) - f ( c )| >
  
  even if | x - c | < "



• Example 4: Show that is not uniformly continuous   over
  
  Note: For any given > 0,   when ,   .
  
  Therefore,   could go to infinity.
  
  Proof : It is very simple by viewing the above "fact".



• Example 5: Show that is uniformly continuous over
  
  Note:
  
  It seems like we need an advance concept to prove the problem above.
  
  ( 1 ) When ,     for  , therefore, the selection of
  
  Could be independent of all c which are larger than or equal to 4.
  
  ( 2 )   is uniformly continuos over [ 0, 4 ], since [0, 4] is a compact set. (This is an advanced concept.)





• Techniques ( Finding the limit of f ( x ) )
  
  ( 1 ) For a polynomial function, f ( x ) or continuous function,
  
  substitution method can be applied to find the limit for f ( x ).
  
  Example :
  
  
  
  Therefore, polynomial function is a continuous function.
  
  ( 2 ) For a rational function,   , substitution method can be applied
  
  to find the limit for f ( x ), unless the following results happen through substitution method.
  
  
  •   : ( indeterminate case )
    
    Usually, there exists common factor ( x - c ) between numerator and denominator that can be
    
    removed before the substitution method can be applied.
    
    ( Actually, there could be any other methods, too. )
    
    Example 1:
    
    
    
    Example 2:
    
    
    
    Example 3:
    
    
    
    ( Substitution method ends with )
  •   : ( the limit does not exist )
    
    ( The behavior of the graph for f ( x ) near x = c can be identified by one - sided limits )
    
    Example :  
    
    



• Special ratio-type function
  
  
  • Try substitute method first, unless the following cases happen.
    
    ( 1 ) : ( indeterminate case )
    
    Example 1 :  
    
    ( Numerator has radical Notation for x. Rationalize the numerator )
    
    
    
    Example 2 :  
    
    ( denominator has radical notation for x. Rationalize the denominator. )
    
    
    
    Example 3 :  
    
        ( i ) L'Hopital rule may be applied.
    
        ( ii ) Sandwich theorem may be applied.
    
    ( 2 ) :   The limit does not exist.
    
    
    Example :  



• Piece-wisely defined function
  
  ( To find the limit for piece-wisely defined function at the separating point in the domain,
  
  the one-side limits may be used. )
  
  Example 1 :  
  
                       
  
  
  Example 2 :  
  
                       



• One-sided limits
  
  
  • Right-sided limit at   x = c.
    
    Given   ,   there exists     such that whenever   ,
    
    ,   then the right sided limit of   f ( x )   at   x = c   and denoted by
    
    .
    
    
    Remarks:
    ( 1 )     means that   x   approaches to   c   from the right side of   c.
    
    ( note that " + " on the right upper corner of c )
    
    ( 2 )   Roughtly speaking,     means that whenever   x   approaches
    
    to   c   from the right side of   c,   the graph point ( x, f ( x ) ) approaches to the point ( c, L )
    
    on the rectangle coordinate place.
  
  
  
  • Left-sided limit at   x = c.
    
    Given   ,   there exists   ,   such that whenever   ,
    
    ,   then the left-sided limit of   f ( x )   at   x = c   is L and denoted by
    
    .
    
    Remarks:
    
    ( 1 )     means that   x   approaches to   c   from the left side of   c
    
    ( note that " - " on the right upper corner of   c  )
    
    ( 2 )   Roughly speaking,     means that whenever x approaches to   c
    
    from the left side of of   c,   the graph point   ( x, f ( x ) )   approaches to the point   ( c, L )
    
    on the plane.



• Remarks:
  
  
  • Basically, the analysis had been used before proving the both-sided limit can be applied
    
    to understand the selection for   ,   given   ,   for proving the one-sided limits.
    
    In the procedures of the proofs for one-sided limits, the     for both
    
    sided limit will be replaced by     for right-sided limit and be replaced by
    
      for the left-sided limit.
  
  
  
  • To find the limit for piece-wisely defined function at the separating point,   c,   of the domain,
    
    both one-sided limits can be applied. Actually, the following theorem tells the relationship
    
    between both one-sided limits and both one-sided limit.



• Theorem:
  
  Iff   ,   then  
  
  ( When both one-sided limits are equal, both sided-exists )
  
  Proof:
  
    Let   ,   i.e   given   ,   there exists     such that
  
  whenever ,     ( right-sided limit ).
  
  For the same   ,   there exists     such that whenever  
  
    (   ),     ( left-sided limit ).
  
  Therefore, for the given   ,   let   ,   then whenever   ,
  
  ( i.e     and   ),     i.e   .
  
    Let     i.e given   ,   there exists     such that whenever
  
  ,     .   Therefore, for the given   ,       whenever
  
    ( right-sided limit ) and     whenever  
  
  ( left-sided limit ).



• Remarks:
  
  For finding the limit at   x = c,   we may find both one-sided limits.
  
  ( 1 ) Iff both one-sided limits exist and are equal, then the both-sided limits are equal to one-sided limit.
  
  Example1:
  
  Find  
  
  Solution:
  
    and  
  
  ( 2 ) Iff both one-sided limits exist but are not equal, then the both one-sided limits don't exist.
  
  ( The graph of   f ( x )   at   x = c   has a jump. )
  
  Example2:
  
  Find  
  
  Solution:
  
  Since   ,   we have,
  
      and  
  
  ( 3 ) If either     or     does not exist, then  
  
  does not exist.
  
  Example3:
  
  Find  
  
  Solution:
  
      and  
  
  Sign Chart:
  
  
  
  
• Definition: