Calculus 3208 
                                Limits and Continuity (7) 
                              Unit 2: Chapter # 1 (Essential Calculus) 
                       Evaluating Limits by Simplifying Rational Expressions 
                                                   
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         Calculating Limits 
           •   Limits with complex fractions resulting in indeterminate forms from substitution 
           •   Simplifying rational expressions, to create expressions with factors that cancel 
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     Limits with Complex Fractions 
     Sometimes a limit involving a complex fraction (fractions within a fraction) yields an indeterminate form  
     from direct substitution.  
     The basic technique used to evaluate such limits is to first simplify the complex fraction as much as possible, then apply 
     any of our other techniques (factoring, conjugates, etc.) as necessary. 
                          1    1
     Consider the limit:                        Simplifying first: 
                         x13
                     lim
                      x2  x2
       Using Direct Substitution: 
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     A Complex Fraction with a Radical 
          Here we’ll need to combine some of the techniques we’ve learned. 
                                1    1
              Consider:          x  2
                           lim
                            x4  x4
         Attack the Fraction First: 
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