Not seeing steps between factoring fractionsI'm looking to solve the limit of the following error function as s goes to 0, but I'm failing on factoring things out. My calculator (TI-89) gives me a nice form I can use, but I cannot manipulate things quite right. Can anyone point out what I'm doing wrong/missing?The function starts as s E ( s ) = s s 2 [ 1 − K ( K 1 s + K 2 ) T s 2 + ( K K 1 + 1 ) s + K K 2 ] I multiplied so as to combine the two fractions: 1 s ⋅ T s + K K 1 + 1 + K K 2 s T s + K K 1 + 1 + K K 2 s And I'm left with T s + K K 1 + 1 + K K 2 s − K K 1 s − K K 2 T s 2 + ( K K 1 + 1 ) s + K K 2 Unfortunately, I can't figure how to simplify out the numerator.The calculator states the answer is T s + 1 T s 2 + ( K K 1 + 1 ) s + K K 2 I can deal with that, as the limit will simply be 1 K K 2 . But what are the steps in between? I don't know how to get rid of the K K x terms in the numerator as I need to.Thanks in advance.

Question

Not seeing steps between factoring fractionsI&#039;m looking to solve the limit of the following error function as s goes to 0, but I&#039;m failing on factoring things out. My calculator (TI-89) gives me a nice form I can use, but I cannot manipulate things quite right. Can anyone point out what I&#039;m doing wrong/missing?The function starts as  s  E  (  s  )  =      s          s      2            [    1    −                  K        (                  K          1                s        +                  K          2                )                    T                  s          2                +        (        K                  K          1                +        1        )        s        +        K                  K          2                      ]  I multiplied so as to combine the two fractions:      1    s    ⋅            T      s      +      K              K        1            +      1      +                        K                      K            2                          s                    T      s      +      K              K        1            +      1      +                        K                      K            2                          s            And I&#039;m left with            T      s      +      K              K        1            +      1      +                        K                      K            2                          s            −      K              K        1            s      −      K              K        2                    T              s        2            +      (      K              K        1            +      1      )      s      +      K              K        2            Unfortunately, I can&#039;t figure how to simplify out the numerator.The calculator states the answer is            T      s      +      1              T              s        2            +      (      K              K        1            +      1      )      s      +      K              K        2            I can deal with that, as the limit will simply be       1          K              K        2            . But what are the steps in between? I don&#039;t know how to get rid of the   K      K    x   terms in the numerator as I need to.Thanks in advance.

lelapem · Accepted Answer

I believe your mistake is somewhere in how you combined the fractions initially. Here&#039;s what the simplification should look like:                    s        E        (        s        )                            =                  s                      s            2                                    [          1          −                                    K              (                              K                1                            s              +                              K                2                            )                                      T                              s                2                            +              (              K                              K                1                            +              1              )              s              +              K                              K                2                                              ]                                                  =                  1          s                −                              K            (                          K              1                        s            +                          K              2                        )                                s            (            T                          s              2                        +            (            K                          K              1                        +            1            )            s            +            K                          K              2                        )                                                            =                              T                          s              2                        +            (            K                          K              1                        +            1            )            s            +            K                          K              2                        −            K            (                          K              1                        s            +                          K              2                        )                                s            (            T                          s              2                        +            (            K                          K              1                        +            1            )            s            +            K                          K              2                        )                                                            =                              T                          s                              2                                      +            s                                s            (            T                          s              2                        +            (            K                          K              1                        +            1            )            s            +            K                          K              2                        )                                                            =                              T            s            +            1                                T                          s              2                        +            (            K                          K              1                        +            1            )            s            +            K                          K              2