Math131 Calculus I                               The Limit Laws                                  Notes 2.3 
             
            I.      The Limit Laws 
                    Assumptions: c is a constant and lim f (x) and limg(x)exist 
                                                         x→a             x→a
                     
                                Limit Law in symbols                                      Limit Law in words 
            1        lim[ f (x) + g(x)] = lim f (x) + lim g(x)                       The limit of a sum is equal to 
                      x→a                   x→a         x→a                              the sum of the limits. 
            2        lim[ f (x) − g(x)] = lim f (x) −lim g(x)                    The limit of a difference is equal to 
                      x→a                   x→a         x→a                           the difference of the limits. 
            3                  limcf (x) = clim f (x)                      The limit of a constant times a function is equal 
                               x→a             x→a                          to the constant times the limit of the function. 
            4          lim[ f (x)g(x)] = lim f (x)⋅lim g(x)]                       The limit of a product is equal to 
                       x→a                 x→a        x→a                              the product of the limits. 
                           f (x)   lim f (x)                                      The limit of a quotient is equal to 
            5         lim        = x→a                  (        ) 
                      x→a                           if lim g(x) ≠ 0                    the quotient of the limits. 
                          g(x)     limg(x)            x→a
                                    x→a
            6                lim[ f (x)]n = [lim f (x)]n                             where n is a positive integer 
                              x→a             x→a
            7                          limc = c                                The limit of a constant function is equal  
                                       x→a                                                  to the constant. 
            8                         limx = a                                  The limit of a linear function is equal 
                                       x→a                                         to the number x is approaching. 
            9                        limxn = an                                      where n is a positive integer 
                                     x→a
           10                       limn x = n a                             where n is a positive integer & if n is even,  
                                     x→a                                                 we assume that a > 0 
                                  n                                          where n is a positive integer & if n is even,  
           11                 lim    f (x) = n lim f (x)                             we assume that             > 0 
                              x→a              x→a                                                     lim f (x)
                                                                                                       x→a
             
            Direct Substitution Property:           If f is a polynomial or rational function and a is in the domain of f,  
                                                    then lim f (x) =  
                                                          x→a
             
             
            “Simpler Function Property”:            If  f (x) = g(x)  when  x ≠ athenlim f (x) = lim g(x), as long as the 
                                                                                        x→a          x→a
                                                    limit exists. 
               Math131 Calculus I                                           Notes 2.3                                                               page 2 
                
               ex#1  Givenlim f(x) = 2,limg(x)=−1, limh(x) =3 use the Limit Laws find lim                                      f (x)h(x)− x2g(x)  
                                  x→3              x→3               x→3                                                x→3
                
                
                
                
                
                
                
                                              2x2 +1
               ex#2  Evaluate  lim                        , if it exists, by using the Limit Laws. 
                                       x→2 x2 +6x−4
                
                
                
                
                
                
                
                
               ex#3  Evaluate:                lim2x2 +3x−5 
                                              x→1
                
                
                
                
                                                   1−(1−x)2
               ex#4  Evaluate:                lim                  
                                              x→0         x
                
                
                
                
                
                
               ex#5  Evaluate:                lim h+4−2 
                                              h→0        h
               Math131 Calculus I                                           Notes 2.3                                                               page 3 
                
               Two Interesting Functions 
                          
               1.        Absolute Value Function  
                
                         Definition:   x =    x  if  x ≥ 0                                                                     
                                                 −x  if  x < 0
                                                 

                                    
                         Geometrically:                 The absolute value of a number indicates its distance from another number. 
                
                          x−c =a   means the number x is exactly _____ units away from the number _____. 
                
                          x−c